TLR8 Agonist VTX-2337 and Pegylated Liposomal Doxorubicin Hydrochloride or Paclitaxel in Treating Patients With Recurrent or Persistent Ovarian Epithelial, Fallopian Tube, or Peritoneal Cavity 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 with the following histologic cell types are eligible: * Serous adenocarcinoma * Endometrioid adenocarcinoma * Mucinous adenocarcinoma * Undifferentiated carcinoma * Clear cell adenocarcinoma * Mixed epithelial adenocarcinoma * Transitional cell carcinoma * Malignant Brenner tumor * Adenocarcinoma not otherwise specified (N.O.S.) * Patient must have measurable disease or detectable (non-measurable) disease: * Measurable disease will be defined by RECIST 1.1 * Measurable disease is defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded) * Each lesion must be ≥ 10 mm when measured by CT, MRI, or caliper measurement by clinical exam or ≥ 20 mm when measured by chest x-ray * Lymph nodes must be ≥ 15 mm in short axis when measured by CT or MRI * Detectable (non-measurable) disease is defined as not having measurable disease but has at least one of the following conditions: * Baseline values of CA-125 ≥ 2 x ULN * Patients whose CA-125 is \< 100 U/mL must undergo a second confirmatory value within a period of ≤ 4 weeks * Patients with a level ≥ 100 U/mL may be entered without confirmatory measurement * The CA-125 assessment for eligibility must be done at least 4 weeks after paracentesis or other surgical procedures * Solid and/or cystic abnormalities on radiographic imaging that do not meet RECIST 1.1 definitions for target lesions * Patient with measurable disease 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 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 * For patients enrolled in Regimen 2, weekly paclitaxel, treatment for their primary disease must have included paclitaxel; this can have been given weekly or every 3 weeks; patients who experienced severe hypersensitivity to paclitaxel are excluded * 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 * For patients enrolled in Regimen 1 which includes PLD, prior treatment with PLD is NOT allowed * For patients enrolled in Regimen 2 which includes weekly paclitaxel, prior treatment with weekly paclitaxel for recurrent or persistent disease is NOT allowed * Patients are allowed to receive, but are not required to receive, non-cytotoxic therapy (such as bevacizumab) as part of their primary treatment regimen; patients are allowed to receive, but are not required to receive, 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 * No patients with history or evidence upon physical examination of 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 of the first date of treatment on this study * No patients who have received oral or parenteral corticosteroids within 2 weeks of study entry or who require ongoing systemic immunosuppressive therapy for any reason * Patients must have a GOG performance status of 0 or 1 * Absolute neutrophil count (ANC) ≥ 1,500/mm\^3; this ANC cannot have been induced or supported by granulocyte colony-stimulating factors * Platelets ≥ 100,000/mm\^3 * Hemoglobin ≥ 9 g/dL * Creatinine ≤ 1.5 x institutional upper limit normal (ULN) * Bilirubin ≤ 1.5 x ULN * SGOT (AST) ≤ 3.0 x ULN * Alkaline phosphatase ≤ 2.5 x ULN * Neuropathy (sensory and motor) less than or equal to grade 1 * Not pregnant or nursing * Negative pregnancy test * Patients of childbearing potential must be practicing an effective form of contraception * Patients should be free of active infection requiring parenteral antibiotics * No patients with active autoimmune disease * "Active" refers to any condition currently requiring therapy * Examples of autoimmune disease include systemic lupus erythematosus, multiple sclerosis, inflammatory bowel disease, and rheumatoid arthritis * Patients with a history of other invasive malignancies, with the exception of non-melanoma skin cancer or localized cancer of the breast, head and neck, or skin, are excluded if there is any evidence of other malignancy being present within the last three years * No patients with known chronic or active hepatitis * No patients with clinically significant cardiovascular disease, this includes: * Uncontrolled hypertension, defined as systolic \> 140 mm Hg or diastolic \> 90 mm Hg * Myocardial infarction or unstable angina within 6 months of the first date of treatment on this study * History of serious ventricular arrhythmia (i.e., ventricular tachycardia or ventricular fibrillation) or cardiac arrhythmias requiring anti-arrhythmic medications (except for atrial fibrillation that is well controlled with anti-arrhythmic medication) * QTc interval ≥ 450 ms on baseline ECG * Baseline ejection fraction ≤ 50% as assessed by echocardiogram or MUGA * New York Heart Association (NYHA) class II or higher congestive heart failure * Grade 2 or higher peripheral ischemia (brief \[\< 24 hours\] episode of ischemia managed non-surgically and without permanent deficit) * No patients with clinical symptoms or signs of gastrointestinal obstruction and who require parenteral hydration or nutrition * Any hormonal therapy directed at the malignant tumor must be discontinued at least one week prior to registration * Continuation of hormone replacement therapy is permitted * No patients who have had previous treatment with VTX-2337, pegylated liposomal doxorubicin, doxorubicin, or any other anthracycline * Patients are excluded if their previous cancer treatment contraindicates this protocol therapy * 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 (prior treatment with pegylated liposomal doxorubicin is NOT allowed) * Patients are allowed to receive, but are not required to receive, non-cytotoxic therapy (such as bevacizumab) as part of their primary treatment regimen * Patients are allowed to receive, but are not required to receive, non-cytotoxic therapy for management of recurrent or persistent disease * Patients must have recovered from effects of recent surgery, radiotherapy, or chemotherapy * 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 that the patient remains free of recurrent or metastatic disease * No patients who have received an investigational agent within 30 days of study entry * Any other prior therapy directed at the malignant tumor, including biological and immunologic agents, must be discontinued at least three weeks prior to registration