Bevacizumab With or Without Everolimus in Treating Patients With Recurrent or Persistent Ovarian Epithelial Cancer, Fallopian Tube Cancer, or Primary Peritoneal Cancer

Inclusion Criteria: * Patients must have recurrent or persistent epithelial ovarian, fallopian tube, or primary peritoneal carcinoma; histologic documentation of the original primary tumor is required via the pathology report * Patients must have measurable disease or detectable (non-measurable) disease: * Measurable disease is defined as at least one lesion that can be accurately measured in at least one dimension (longest dimension to be recorded); each lesion must be greater than or equal to 10 mm when measured by computed tomography (CT), magnetic resonance imaging (MRI) or caliper measurement by clinical exam; or greater than or equal to 20 mm when measured by chest x-ray; lymph nodes must be greater than or equal to 15 mm in short axis when measured by CT or MRI * Detectable disease in a patient is defined as one who does not have measurable disease but has at least one of the following conditions: * Baseline values of CA-125 at least 2 x upper limit of normal (ULN) * Ascites and/or pleural effusion attributed to tumor * Solid and/or cystic abnormalities on radiographic imaging that do not meet RECIST 1.1 definitions for target lesions * Patients in the measurable disease cohort must have at least one ?target lesion? to be used to assess response on this protocol as defined by RECIST 1.1; tumors within a previously irradiated field will be designated as ?non-target? lesions unless progression is documented or a biopsy is obtained to confirm persistence at least 90 days following completion of radiation therapy * Patients must not be eligible for a higher priority Gynecologic Oncology Group (GOG) protocol, if one exists; in general, this would refer to any active GOG phase III protocol or Rare Tumor Protocol for the same patient population * Patients who have had one prior treatment must have a GOG performance status of 0, 1, or 2; patients who have had two or three prior treatments must have a GOG performance status of 0 or 1 * Patients should be free of active infection requiring parenteral antibiotics (with the exception of uncomplicated urinary tract infection \[UTI\]) * Recovery from effects of recent surgery, radiotherapy, or chemotherapy * Any hormonal therapy directed at the malignant tumor must be discontinued at least one week prior to registration * Any other prior therapy directed at the malignant tumor, including biological and immunologic agents, must be discontinued at least 3 weeks prior to registration (including small molecules and murine monoclonal antibodies); chimeric or human or humanized monoclonal antibodies (including bevacizumab) or vascular endothelial growth factor (VEGF) receptor fusion protein (including VEGF Trap, aflibercept) must be discontinued for at least 12 weeks prior to registration; no investigational therapy within 30 days prior to the first date of study treatment * Prior therapy: * Patients must have had one prior platinum-based chemotherapeutic regimen for management of primary disease containing carboplatin, cisplatin, or another organoplatinum compound' this initial treatment may have included intraperitoneal therapy, consolidation, non-cytotoxic agents or extended therapy administered after surgical or non-surgical assessment * Patients are allowed to receive, but are not required to receive, two additional cytotoxic regimens for management of recurrent or persistent disease, with no more than 1 non-platinum, non-taxane regimen * Patients are allowed to receive, but are not required to receive, biologic (non-cytotoxic) therapy as part of their primary treatment regimen; patients must have NOT received any non-cytotoxic therapy for management of recurrent or persistent disease * Patients who have received only one prior cytotoxic regimen (platinum-based regimen for management of primary disease), must have a platinum-free interval of less than 12 months, or have progressed during platinum-based therapy, or have persistent disease after a platinum-based therapy * For the purposes of this study, poly (adenosine diphosphate \[ADP\]-ribose) polymerase (PARP) inhibitors will be considered ?cytotoxic,? and prior treatment with PARP inhibitors for primary or recurrent disease WILL be allowed (either alone or in combination with chemotherapy) * Absolute neutrophil count (ANC) greater than or equal to 1,500/mcl * Platelets greater than or equal to 100,000/mcl * Creatinine less than or equal to 1.5 x institutional upper limit normal (ULN) * Bilirubin less than or equal to 1.5 x ULN * Serum glutamic oxaloacetic transaminase (SGOT) (aspartate aminotransferase \[AST\]) less than or equal to 3 x ULN * Alkaline phosphatase less than or equal to 2.5 x ULN * Prothrombin time (PT) such that international normalized ratio (INR) is less than or equal to 1.5 x ULN (or an in-range INR, usually between 2 and 3, if a patient is on a stable dose of therapeutic warfarin for management of deep vein thrombosis including pulmonary embolism) * Partial thromboplastin time (PTT) less than or equal to 1.5 times the upper limit of normal * Fasting serum cholesterol less than or equal to 300 mg/dL OR less than or equal to 7.75 mmol/L AND * Fasting triglycerides less than or equal to 300 mg/dL or 3.42 mmol/L * Urine protein creatinine (UPC) ratio must be \< 1.0 gm; if UPC ratio \>= 1, collection of 24-hour urine measurement of urine protein is recommended; 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 * Patients of childbearing potential must have a negative serum pregnancy test prior to the study entry and be practicing an effective form of contraception * Patients must have signed an approved informed consent and authorization permitting the release of personal health information * Patients must meet pre-entry requirements * Patients with clinically significant cardiovascular disease; this includes: * Uncontrolled hypertension, defined as systolic \> 150 mm Hg or diastolic \> 90 mm Hg * Myocardial infarction or unstable angina within 6 months prior to registration * New York Heart Association (NYHA) class II or greater congestive heart failure * Serious cardiac arrhythmia requiring medication; this does not include asymptomatic, atrial fibrillation with controlled ventricular rate * Patients who have received prior treatment with an anthracycline (including doxorubicin and or liposomal doxorubicin) and have an ejection fraction \< 50% * Common Terminology Criteria for Adverse Events (CTCAE) grade 2 or greater peripheral vascular disease (at least brief \[\< 24 hours (hrs)\] episodes of ischemia managed non-surgically and without permanent deficit) Exclusion Criteria: * Patients who have received prior everolimus or any other mammalian target of rapamycin (mTOR) inhibitor * Patients with other invasive malignancies, with the exception of non-melanoma skin cancer and other specific malignancies, are excluded if there is any evidence of other malignancy being present within the last three years; patients are also excluded if their previous cancer treatment contraindicates this protocol therapy * Patients who have received prior radiotherapy to any portion of the abdominal cavity or pelvis OTHER THAN for the treatment of ovarian, fallopian tube or primary peritoneal cancer within the last three years are excluded; prior radiation for localized cancer of the breast, head and neck, or skin is permitted, provided that it was completed more than three years prior to registration, and the patient remains free of recurrent or metastatic disease * Patients who have received prior chemotherapy for any abdominal or pelvic tumor OTHER THAN for the treatment of ovarian, fallopian tube or primary peritoneal cancer within the last three years are excluded; patients may have received prior adjuvant chemotherapy for localized breast cancer, provided that it was completed more than three years prior to registration, and the patient remains free of recurrent or metastatic disease * Patients with serious non-healing wound, ulcer, or bone fracture; this includes history of abdominal fistula, gastrointestinal perforation or intra-abdominal abscess within 28 days prior to the first date of study treatment * Patients with active bleeding or pathologic conditions that carry high risk of bleeding, such as known bleeding disorder, coagulopathy, or tumor involving major vessels * Patients with history or evidence upon physical examination of central nervous system (CNS) disease, including primary brain tumor, seizures not controlled with standard medical therapy, any brain metastases, or history of cerebrovascular accident (CVA, stroke), transient ischemic attack (TIA) or subarachnoid hemorrhage within six months prior to the first date of study treatment * Known hypersensitivity to murine or chimeric antibodies * Major surgery within 28 days prior to the first date of study treatment * Patients with clinical symptoms or signs of gastrointestinal obstruction and patients who require parenteral hydration and/or nutrition * Patients with medical history or conditions not otherwise previously specified which in the opinion of the investigator should exclude participation in this study; the investigator can consult the Study Chair or Study Co-Chairs for uncertainty in this regard * Patients who are pregnant or nursing * Patients with active hepatitis B or C * Laboratory confirmation to exclude hepatitis B and C is required in any patient considered at high risk for hepatitis B or C; risk factors can include: * You currently live in (or have lived in) Asia, Africa, Central and South America, Eastern Europe, Spain, Portugal, and Greece * Known or suspected past hepatitis B infection * Known or suspected past hepatitis C infection (including past interferon ?curative? treatment) * Blood transfusion(s) prior to 1990 * Current or prior IV drug use * Current or prior dialysis * Household contact with hepatitis B or hepatitis C infected person(s) * Current or prior high-risk sexual activity * Body piercing or tattoos * Mother known to have hepatitis B * History suggestive of hepatitis B infection, e.g., dark urine, jaundice, right upper quadrant pain