Cabozantinib-S-Malate in Treating Patients With Recurrent or Progressive Ovarian, Fallopian Tube, or Primary Peritoneal Cancer

Inclusion Criteria: * A retrospective review of all patients entered will be performed to confirm clear cell histology; patients must have recurrent or, progressive clear cell ovarian cancer not solely based on cancer antigen (CA)-125; primary tumors must be at least 50% clear cell histomorphology in order to be eligible or have a histologically documented recurrence with at least 50% clear cell histomorphology; recurrence should be biopsy proven unless the tumor is located in an area deemed unsafe to biopsy by the surgeon; if a biopsy can be obtained without significant risk, then biopsy should be obtained * If the primary tumor had at least 50% clear cell histomorphology, a biopsy of the recurrent or persistent tumor is not required; the percentage of clear cell histomorphology must be documented in the pathology report or in an addendum to the original report; if slides of the primary tumor are not available for review due to disposal of slides by the histology laboratory (typically 10 years after diagnosis), biopsy of recurrent or persistent disease is required * If slides of the primary tumor are not available for review, a biopsy of the recurrent or persistent tumor is required to confirm at least 50% clear cell histomorphology; the percentage of involvement must be documented in the pathology report or in an addendum to the original report * All patients must have measurable disease as defined by Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 * All patients must submit unstained slides of primary or recurrent tumor for translational analysis * Patients must have at least one "target lesion" to be used to assess response on this protocol as defined by RECIST 1.1 * Patients must have had one prior platinum-based chemotherapeutic regimen for management of primary disease; platinum sensitive and resistant patients are eligible * Patients are allowed to receive, but are not required to receive, one additional cytotoxic regimen for management of recurrent or persistent disease * Concomitant use of additional anti-neoplastic agents will not be allowed in this study * Patients may not have received previous therapy with a MET inhibitor * Patients must not be eligible for a higher priority (e.g.; phase II/III), National Surgical Adjuvant Breast and Bowel Project, Radiation Therapy Oncology Group, and Gynecologic Oncology Group (NRG) protocol for the same population if one exists * Patients must be recovered from effects of recent surgery (28 days must elapse between surgery and the start of treatment with cabozantinib) * Patients must have \>= 4 weeks since prior chemotherapy or radiation (\>= 6 weeks for nitrosoureas or mitomycin C) * Appropriate stage for study entry based on the following diagnostic workup: * History/physical examination within 28 days prior to registration * The trial is open only to women with recurrent, progressive clear cell carcinoma of the ovary * Patients must have an Eastern Cooperative Oncology Group (ECOG) performance status of 0, 1, or 2 (Karnofsky \>= 60%) within 28 days prior to registration * Absolute neutrophil count (ANC) \>= 1,500/mcl * Platelets greater than or equal to 100,000/mcl * Leukocytes \>= 3,000/mcL * Hemoglobin \>= 9 g/dL * Serum albumin \>= 2.8 g/dL * Prothrombin time (PT) such that international normalized ratio (INR) is less than or equal to 1.3 x 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 times the ULN OR creatinine clearance \>= 50 mL/min/1.73 m\^2 for patients with creatinine levels above institutional normal * The urine protein: creatinine ratio (UPCR) is derived as follows: protein concentration (mg/dL)/creatinine (mg/dL); patients must have a UPCR \< 1.0 to allow participation in the study * Bilirubin less than or equal to 1.5 ULN * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) less than or equal to 2.5 times the ULN, unless subjects have liver metastasis, in which case both AST and ALT must be less than or equal to 5 times the ULN * Lipase less than or equal to 2 x ULN * No clinical evidence of pancreatitis * Patients must have a normal baseline thyroid stimulating hormone (TSH); a history of hypothyroidism and/or hyperthyroidism is allowed * Women of childbearing potential must have a negative pregnancy test at screening; women of childbearing potential include women who have experienced menarche and who have not undergone successful surgical sterilization (hysterectomy, bilateral tubal ligation, or bilateral oophorectomy) or are not postmenopausal; post-menopause is defined as amenorrhea \>= 12 consecutive months; note: women who have been amenorrheic for 12 or more months are still considered to be of childbearing potential if the amenorrhea is possibly due to prior chemotherapy, anti-estrogens, ovarian suppression or any other reversible reason; women should stop breastfeeding while participating in this trial; male partners of women participants should also use medically-acceptable forms of contraception during the study * The patient must provide study-specific informed consent prior to study entry Exclusion Criteria: * Human immunodeficiency virus (HIV) positive patients * Patients with serious non-healing wound, ulcer, or bone fracture * Patients with active bleeding or pathologic conditions that carry high risk of bleeding, such as known bleeding disorder, coagulopathy, or tumor involving major vessels * The subject requires concomitant treatment, in therapeutic doses, with anticoagulants such as warfarin or warfarin-related agents, heparin, thrombin or factor Xa inhibitors, or antiplatelet agents (e.g., clopidogrel); low dose aspirin (=\< 81 mg/day), low-dose warfarin (=\< 1 mg/day), and prophylactic low molecular weight heparin (LMWH) are permitted * 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 6 months of the first date of treatment on this study * Patients with clinically significant cardiovascular disease; this includes: * Poorly controlled hypertension (\> 140 mm Hg and \> 90 mm Hg for systolic and diastolic blood pressure \[BP\]) are ineligible * Myocardial infarction or unstable angina within 6 months prior to registration; New York Heart Association (NYHA) grade II or greater congestive heart failure * Cardiac arrhythmia requiring medication * Grade II or greater peripheral vascular disease based on National Cancer Institute (NCI) Common Toxicity Criteria (CTC); e.g. ischemic rest pain, minor tissue loss, and ulceration or gangrene * Patients with a pre-existing thyroid abnormality who are unable to maintain thyroid function in the normal range with medication are ineligible; patients with a history of hypothyroidism are eligible provided they are currently euthyroid * Patients who have a major surgical procedure, or significant traumatic injury within 28 days prior to the first date of treatment on this study, or anticipation of need for major surgical procedure during the course of the study; patients with placement of vascular access device or core biopsy within 7 days prior to the first date of treatment on this study * Patients with other invasive malignancies, with the exception of non-melanoma skin cancer, who had (or have) any evidence of other cancer present within the last 5 years or whose previous cancer treatment contraindicates this protocol therapy * Patients must not be eligible for a higher priority NRG clear cell protocol (Gynecologic Oncology Group \[GOG\]-0283); rare patients ineligible for GOG-0283 may be eligible for this trial without prior treatment on dasatinib therapy (e.g. they have been deemed allergic to dasatinib) * Patients who cannot swallow pills