Pegylated Liposomal Doxorubicin Hydrochloride With Atezolizumab and/or Bevacizumab in Treating Patients With Recurrent Ovarian, Fallopian Tube, or Primary Peritoneal Cancer

Inclusion Criteria: * Patients must have the psychological ability and general health that permits completion of the study requirements and required follow up * Administration of study drugs (pegylated liposomal doxorubicin, bevacizumab, atezolizumab) may have an adverse effect on pregnancy and poses a risk to the human fetus, including embryo-lethality; women of child-bearing potential (WOCBP) must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation, and for 5 months (150 days) after the last dose of study agent; should a woman become pregnant or suspect she is pregnant while she is participating in this study, she should inform her treating physician immediately. (06/29/2017) * Submission of tumor tissue is required for all patients; investigators should check with their site pathology department regarding release of biospecimens before approaching patients about participation in the trial * High grade ovarian cancer, including high grade serous; clear cell; endometrioid, grade 3; and others (adenocarcinoma, not otherwise specified \[NOS\]; mixed epithelial carcinoma; undifferentiated carcinoma); NOTE: low grade serous, mucinous and carcinosarcoma histologies are excluded due to their different underlying genomic features and/or clinical behavior; ovarian cancer = ovarian, fallopian tube or primary peritoneal cancer; required data element: submission of pathology report * Recurrent, platinum resistant ovarian cancer (defined as progression within \< 6 months from completion of platinum based therapy; the date should be calculated from the last administered dose of platinum therapy) * 1-2 prior regimens (including primary therapy); hormonal therapies (e.g., tamoxifen, aromatase inhibitors) will not count toward the prior regimen limit; PARP inhibitors given in the maintenance setting post response to platinum-based therapy will not count as a separate regimen from the preceding platinum-based therapy. (30-OCT-2020) * Measurable disease (defined by RECIST v1.1) or evaluable disease (defined as solid and/or cystic abnormalities on radiographic imaging that do not meet RECIST 1.1 definitions for target lesions OR ascites and/or pleural effusion that has been pathologically demonstrated to be disease related in the setting of cancer antigen \[CA\] 125 \>= 2 x upper limit of normal \[ULN\]) * Age \>= 18 * Performance status 0, 1 or 2 * Absolute neutrophil count (ANC) \>= 1,500/mcl (within 14 days prior to registration) * Platelets \>= 100,000/mcl (within 14 days prior to registration) * Hemoglobin (Hgb) \>= 8 g/dl (within 14 days prior to registration) * Creatinine =\< 1.5 x institutional upper limit of normal (ULN) (within 14 days prior to registration) * Urine protein creatinine (UPC) ratio must be \< 1.0 (within 14 days prior to registration); if UPC ratio \>= 1, collection of 24-hour urine measurement of urine protein is recommended (24-hour urine protein level must be \< 1000 mg for patient enrollment); If UPC ratio cannot be calculated because the urine protein is below the lower limit of detection of the assay this will not exclude the patient (10/22/2018) (30-OCT-2020); UPC ratio of spot urine is an estimation of the 24-hour urine protein excretion - a UPC ratio of 1 is roughly equivalent to a 24-hour urine protein of 1 gm * Total bilirubin =\< 1.5 x ULN (patients with known Gilbert disease who have serum bilirubin level =\< 3 x ULN may be enrolled) (within 14 days prior to registration) * Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) =\< 3 x ULN (AST and/or ALT =\< 5 x ULN for patients with liver involvement) (within 14 days prior to registration) * International normalized ratio (INR) and activated partial thromboplastin time (aPTT) =\< 1.5 x ULN (or on stable dose of therapeutic anticoagulation, such as low-molecular-weight heparin, warfarin or rivaroxaban) (10/16/2017) * Thyroid-stimulating hormone (TSH) within normal limits (Euthyroid patients on thyroid replacement therapy allowed provided TSH \< ULN) (02/20/2019) * The patient or legally authorized representative must provide study-specific informed consent prior to study entry and, for patients treated in the United States (U.S.), authorization permitting release of personal health information Exclusion Criteria: * Patients with prior allogeneic bone marrow transplantation or prior solid organ transplantation * Patients who have had systemic anticancer therapy (e.g., chemotherapy, targeted therapy including PARP inhibitors or bevacizumab) within 3 weeks prior to entering the study (30-OCT-2020) * Patients who have had hormonal therapy (e.g., tamoxifen, aromatase inhibitor) within 1 week prior to entering the study * Patients with prior treatment with anti-programmed cell death (PD)-1, anti- programmed cell death ligand (PD-L)1 or anti-cytotoxic T-lymphocyte-associated protein (CTLA)-4 therapeutic antibody or other similar agents (10/16/2017) * Patients with prior treatment with bevacizumab (or any other anti vascular therapy, e.g., cediranib) for platinum resistant recurrence; (Note: prior bevacizumab in initial therapy and/or platinum sensitive recurrent setting is allowed) (10/16/2017) * Patients with prior treatment with PLD * Prior radiotherapy to the abdomen or pelvis * Patients who have not recovered from adverse events to =\< grade 1 (other than alopecia) due to agents administered more than 3 weeks earlier (10/16/2017); however, the following therapies are allowed: * Hormone replacement therapy or oral contraceptives * Herbal therapy \> 1 week prior to cycle 1, day 1 (herbal therapy intended as anticancer therapy must be discontinued at least 1 week prior to cycle 1, day 1) * Palliative radiotherapy for bone metastases \> 2 weeks prior to cycle 1, day 1 * Treatment with any other investigational agent within 4 weeks prior to cycle 1, day 1 * Treatment with systemic immunostimulatory agents (including, but not limited to, interferon \[IFN\]-alpha or interleukin \[IL\]-2) within 6 weeks prior to cycle 1, day 1 * Treatment with systemic immunosuppressive medications (including, but not limited to, prednisone, cyclophosphamide, azathioprine, methotrexate, thalidomide, and anti-tumor necrosis factor \[anti-TNF\] agents) within 2 weeks prior to cycle 1, day 1 * Patients who have received acute, low dose, systemic immunosuppressant medications (e.g., a one-time dose of dexamethasone for nausea or steroids as computed tomography \[CT\] scan contrast premedication) may be enrolled * The use of inhaled corticosteroids and mineralocorticoids (e.g., fludrocortisone) for patients with orthostatic hypotension or adrenocortical insufficiency is allowed * Patients taking bisphosphonate therapy for symptomatic hypercalcemia within the past 28 days; use of bisphosphonate therapy for other reasons (e.g., bone metastasis or osteoporosis) is allowed * Patients with known primary central nervous system (CNS) malignancy or symptomatic CNS metastases are excluded, with the following exceptions: * Patients with asymptomatic untreated CNS disease may be enrolled, provided all of the following criteria are met: * Evaluable or measurable disease outside the CNS * No metastases to brain stem, midbrain, pons, medulla, cerebellum, or within 10 mm of the optic apparatus (optic nerves and chiasm) * No history of intracranial hemorrhage or spinal cord hemorrhage * No ongoing requirement for dexamethasone for CNS disease; patients on a stable dose of anticonvulsants are permitted * No neurosurgical resection or brain biopsy within 28 days prior to cycle 1, day 1 * Patients with asymptomatic treated CNS metastases may be enrolled, provided all the criteria listed above are met as well as the following: * Radiographic demonstration of improvement upon the completion of CNS directed therapy and no evidence of interim progression between the completion of CNS directed therapy and the screening radiographic study * No stereotactic radiation or whole-brain radiation within 28 days prior to cycle 1, day 1 * Screening CNS radiographic study \>= 4 weeks from completion of radiotherapy and \>= 2 weeks from discontinuation of corticosteroids * Known hypersensitivity to Chinese hamster ovary cell products or other recombinant human antibodies * History of severe allergic, anaphylactic, or other hypersensitivity reactions to chimeric or humanized antibodies or fusion proteins * Patients requiring treatment with a receptor activator of nuclear factor kappa-B ligand (RANKL) inhibitor (e.g. denosumab) who cannot discontinue it before treatment with atezolizumab * Known clinically significant liver disease, including active viral, alcoholic, or other hepatitis; cirrhosis; fatty liver; and inherited liver disease * Patients with past or resolved hepatitis B infection (defined as having a negative hepatitis B surface antigen \[HBsAg\] test and a positive anti-HBc \[antibody to hepatitis B core antigen\] antibody test) are eligible * Patients positive for hepatitis C virus (HCV) antibody are eligible only if polymerase chain reaction (PCR) is negative for HCV ribonucleic acid (RNA) * History or risk of autoimmune disease, including but not limited to systemic lupus erythematosus, rheumatoid arthritis, inflammatory bowel disease, vascular thrombosis associated with antiphospholipid syndrome, Wegener's granulomatosis, Sjogren's syndrome, Guillain-Barre syndrome, multiple sclerosis, autoimmune thyroid disease, vasculitis, or glomerulonephritis (02/20/2019) * Patients with a history of autoimmune hypothyroidism on a stable dose of thyroid replacement hormone are eligible * Patients with controlled type 1 diabetes mellitus on a stable insulin regimen, or type 2 diabetes mellitus are eligible * Patients with eczema, psoriasis, lichen simplex chronicus or vitiligo with dermatologic manifestations only (e.g., patients with psoriatic arthritis would be excluded) are permitted provided that they meet the following conditions: * Patients with psoriasis must have a baseline ophthalmologic exam to rule out ocular manifestations * Rash must cover less than 10% of body surface area (BSA) * Disease is well controlled at baseline and only requiring low potency topical steroids (e.g., hydrocortisone 2.5%, hydrocortisone butyrate 0.1%, fluocinolone 0.01%, desonide 0.05%, alclometasone dipropionate 0.05%) * No acute exacerbations of underlying condition within the last 12 months (not requiring psoralen plus ultraviolet A radiation \[PUVA\], methotrexate, retinoids, biologic agents, oral calcineurin inhibitors; high potency or oral steroids) * History of idiopathic pulmonary fibrosis, pneumonitis (including drug induced), organizing pneumonia (i.e., bronchiolitis obliterans, cryptogenic organizing pneumonia, etc.), or evidence of active pneumonitis on screening chest computed tomography (CT) scan; history of radiation pneumonitis in the radiation field (fibrosis) is permitted * Patients with active tuberculosis (TB) are excluded * Severe infections within 4 weeks prior to cycle 1, day 1, including but not limited to hospitalization for complications of infection, bacteremia, or severe pneumonia * Signs or symptoms of infection within 2 weeks prior to cycle 1, day 1 * Received oral or intravenous (IV) antibiotics within 2 weeks prior to cycle 1, day 1; patients receiving prophylactic antibiotics (e.g., for prevention of a urinary tract infection or chronic obstructive pulmonary disease) are eligible * Major surgical procedure within 28 days prior to cycle 1, day 1 or anticipation of need for a major surgical procedure during the course of the study * Administration of a live, attenuated vaccine within 4 weeks before cycle 1, day 1 or anticipation that such a live, attenuated vaccine will be required during the study and up to 5 months after the last dose of atezolizumab (06/29/2017) * Influenza vaccination should be given during influenza season only (approximately October to March); patients must not receive live, attenuated influenza vaccine within 4 weeks prior to cycle 1, day 1 or at any time during the study * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements * Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial (08-JAN-2021) * Prior invasive malignancy (except non-melanomatous skin cancer) unless disease free for a minimum of 3 years, with the exception of those with a negligible risk of metastases or death, such as carcinoma in situ of the breast or cervix * Severe, active co-morbidity defined as follows: * Current (within 28 days of cycle 1, day 1) signs and/or symptoms of bowel obstruction * Patients who require parental hydration and/or nutrition * Patients who require drainage gastrostomy tube * Evidence of bleeding diathesis or clinically significant coagulopathy * Serious, non-healing or dehiscing wound, active ulcer or untreated bone fracture * History of hemoptysis (\>= 1/2 teaspoon of bright red blood per episode) within 1 month of study enrollment * Significant cardiovascular or cerebrovascular disease including: * Uncontrolled hypertension (systolic blood pressure \[SBP\] \>= 150 and/or diastolic blood pressure \[DBP\] \>= 90) * History of myocardial infarction within 6 months * Unstable angina * New York Heart Association functional classification II, III or IV * Baseline ejection fraction =\< 50% as assessed by echocardiogram or multi-gated acquisition (MUGA) * Cerebral vascular accident (CVA) or transient ischemic attack (TIA) within 6 months * Significant vascular disease (e.g., aortic aneurysm, requiring surgical repair or peripheral arterial thrombosis) within 6 months * History of abdominal/pelvic or tracheoesophageal fistula or gastrointestinal perforation and/or abdominal/pelvic abscess within 6 months prior to initiation of treatment (02/20/2019) * Pregnant or lactating patients