Cabozantinib or Paclitaxel in Treating Patients With Persistent or Recurrent Epithelial Ovarian, Fallopian Tube, or Primary 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 must have measurable disease or non-measurable (detectable) disease: * 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 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 * Non-measurable (detectable) disease is defined in this protocol as the absence of measurable disease but at least one of the following conditions: * 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 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 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 or rare tumor protocol for the same patient population; in addition, patients must not be eligible for the currently active phase II cytotoxic protocol in platinum resistant disease * Patients must have a GOG performance status of 0, 1, or 2 * Recovery from effects of recent surgery, radiotherapy, or chemotherapy to baseline or CTCAE =\< grade 1 toxicity from all prior therapies except alopecia and other non-clinically significant adverse events (AE's): * Patients should be free of active infection requiring antibiotics (with the exception of uncomplicated urinary tract infection \[UTI\]) * Any hormonal therapy directed at the malignant tumor must be discontinued at least one week prior to treatment * Any other prior therapy directed at the malignant tumor, including chemotherapy, biological/targeted (non-cytotoxic) agents and immunologic agents, must be discontinued at least three weeks prior to treatment * Chimeric or human or humanized monoclonal antibodies (including bevacizumab) or vascular endothelial growth factor (VEGF) receptor fusion proteins (including VEGF TRAP/aflibercept) must be discontinued for at least 12 weeks prior to treatment * Investigational agents must be discontinued for at least 28 days prior to treatment * Any prior radiation therapy must be discontinued at least four weeks prior to 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, biologic/targeted (non-cytotoxic) agents (e.g., bevacizumab) or extended therapy administered after surgical or non-surgical assessment; if patients were treated with paclitaxel for their primary disease, this could have been given weekly or every 3 weeks * 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; treatment with weekly paclitaxel for recurrent or persistent disease is NOT allowed * Patients are allowed to receive, but are not required to receive, biologic/targeted (non-cytotoxic) therapy as part of their primary treatment regimen * Patients must have NOT received any biologic/targeted (non-cytotoxic) therapy targeting the VEGF and/or MET pathways for management of recurrent or persistent disease * For the purposes of this study, poly (adenosine diphosphate \[ADP\]-ribose) polymerase (PARP) inhibitors will be considered "cytotoxic"; patients are allowed to receive, but are not required to receive, PARP inhibitors for management of primary or recurrent/persistent disease (either alone or in combination with cytotoxic chemotherapy); PARP inhibitors will NOT count as a prior regimen when given alone * Absolute neutrophil count (ANC) greater than or equal to 1,500/mcl * Platelets greater than or equal to 100,000/mcl * Hemoglobin greater than or equal to 9 g/dL * Prothrombin time (PT) such that international normalized ratio (INR) is less than or equal to 1.3 x institutional upper limit of normal (ULN) * Partial thromboplastin time (PTT) less than or equal to 1.3 x ULN * Creatinine less than or equal to 1.5 x ULN * Phosphorus, corrected calcium, magnesium and potassium greater than or equal to institutional lower limit of normal (LLN) * 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 * Bilirubin less than or equal to 1.5 x ULN * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) less than or equal to 3 x ULN * Alkaline phosphatase less than or equal to 2.5 x ULN * Albumin greater than or equal to 2.8 g/dL * Lipase less than or equal to 2 x ULN * No radiologic or clinical evidence of pancreatitis * Patients must have a normal baseline thyroid stimulating hormone (TSH); a history of hypothyroidism and/or hyperthyroidism is allowed * Neuropathy (sensory and motor) less than or equal to grade 1 * Patients of childbearing potential must have a negative pregnancy test prior to the study entry and be practicing an effective form of contraception; sexually active subjects must agree to use medically accepted barrier methods of contraception (e.g., male or female condom) during the course of the study and for 4 months after the last dose of study drug, even if oral contraceptives are also used; all subjects of reproductive potential must agree to use both a barrier method and a second method of birth control during the course of the study and for 4 months after the last dose of study drug; pregnant women are excluded from this study * Patients must have signed an approved informed consent and authorization permitting the release of personal health information * Patients must meet pre-entry requirements Exclusion Criteria: * Patients who have had previous treatment with cabozantinib; patients who have received previous treatment with weekly paclitaxel for recurrent or persistent disease * Patients with a history of 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 or thoracic cavity 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 * Uncontrolled hypertension, defined as systolic greater than 140 mm Hg or diastolic greater than 90 mm Hg despite antihypertensive medications * Myocardial infarction or unstable angina within 6 months prior to registration * New York Heart Association (NYHA) class II or greater congestive heart failure * History of serious ventricular arrhythmia (i.e., ventricular tachycardia or ventricular fibrillation) or serious cardiac arrhythmia requiring medication; this does not include asymptomatic atrial fibrillation with controlled ventricular rate * Any history of congenital long QT syndrome * The subject has a corrected QT interval calculated by the Fridericia formula (QTcF) \> 500 ms within 28 days before randomization; note: if initial QTcF is found to be \> 500 ms, two additional electrocardiogram (EKGs) separated by at least 3 minutes should be performed; if the average of these three consecutive results for QTcF is =\< 500 ms, the subject meets eligibility in this regard * Patients with serious non-healing wound, ulcer, or bone fracture within 28 days before treatment * Patients with history of organ transplant * Patients with active bleeding or pathologic conditions that carry high risk of bleeding, such as known bleeding disorder, coagulopathy, or tumor involving (in contact with, invading or encasing) major vessels * Patients who have experienced any of the following: * Clinically-significant gastrointestinal bleeding within 6 months before the first dose of study treatment * Hemoptysis of \>= 0.5 teaspoon (2.5 mL) of red blood within 3 months before the first dose of study treatment * Any other signs indicative of pulmonary hemorrhage within 3 months before the first dose of study treatment * Patients who have radiographic evidence of cavitating pulmonary lesion(s) * Patients who have tumor invading the gastrointestinal (GI) tract (esophagus, stomach, small or large bowel, rectum or anus), or any evidence of endotracheal or endobronchial tumor within 28 days before treatment * Gastrointestinal disorders, particularly those with potential risk of perforation or fistula formation including: * Any of the following within 28 days of registration * Intra-abdominal tumor/metastases invading GI mucosa * Active peptic ulcer disease * Inflammatory bowel disease (including ulcerative colitis and Crohn's disease), diverticulitis, cholecystitis, symptomatic cholangitis or appendicitis * Malabsorption syndrome * Any of the following within 6 months of registration * Abdominal fistula * Gastrointestinal perforation * Bowel obstruction or gastric outlet obstruction; note: patients requiring drainage gastrostomy (e.g., percutaneous endoscopic gastrostomy \[PEG\] tube) and/or parenteral hydration and/or nutrition are not eligible * Intra-abdominal abscess; note: complete resolution of an intra-abdominal abscess must be confirmed prior to registration even if the abscess occurred more than 6 months prior to registration * Patients with history or evidence upon physical examination of central nervous system (CNS) disease, including primary brain tumor, seizures which are not controlled with non-enzyme inducing anticonvulsants, any brain metastases and/or epidural disease, 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 * The subject requires chronic concomitant treatment of strong cytochrome P450, family 3, subfamily A, polypeptide 4 (CYP3A4) inducers (e.g., dexamethasone, phenytoin, carbamazepine, rifampin, rifabutin, rifapentine, phenobarbital, and St. John's wort) * Patients who are unable or unwilling to swallow tablets * Patients who are pregnant or nursing * The subject requires concomitant treatment, in therapeutic doses, with anti-coagulants such as warfarin or warfarin-related agents, heparin, thrombin or factor Xa inhibitors or antiplatelet agents (i.e. clopidogrel); low dose aspirin (=\< 81 mg/day) low-dose warfarin (=\< 1 mg/day) and prophylactic low molecular weight heparin are permitted * Major surgery within 3 months of the first dose of cabozantinib if there were no wound healing complications or within 6 months of the first dose of cabozantinib if there were wound complications * Minor surgery within 1 month of the first dose of cabozantinib if there were no wound healing complications or within 3 months of the first dose of cabozantinib if there were wound complications * Patients with concurrent uncompensated hypothyroidism or thyroid dysfunction within 7 days before the first dose of study treatment