Paclitaxel With or Without Pazopanib Hydrochloride in Treating Patients With Persistent or Recurrent 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 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 in a patient is defined in this protocol as one who does not have measurable disease but has 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 Gynecology 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 who have received one prior regimen must have a GOG performance status of 0, 1, or 2 * Patients who have received two or three prior regimens must have a GOG performance status of 0 or 1 * Recovery from effects of recent surgery, radiotherapy, or chemotherapy * 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 registration * 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 registration; 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 registration * At least 4 weeks must have elapsed since the patient underwent any major surgery (e.g., major: laparotomy, laparoscopy, thoracotomy, video-assisted thorascopic surgery \[VATS\]); there is no restriction on minor procedures (e.g., minor: central venous access catheter placement, ureteral stent placement or exchange, paracentesis, thoracentesis) * 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 can 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 platelet-derived growth factor (PDGF) pathways for management of recurrent or persistent disease * For the purposes of this study, poly (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 1.5 x upper limit of normal (ULN) (or an in-range INR, usually between 2 and 3, if a patient is on a stable dose of therapeutic warfarin) * Partial thromboplastin time (PTT) less than or equal to 1.5 x ULN * Creatinine less than or equal to 1.5 x institutional ULN * Urine protein should be screened by urinalysis; if urine protein is 2+ or higher, 24-hour urine protein should be obtained and the level must be \< 1000 mg (\< 1.0 g/24 hours \[hrs\]) for patient enrollment * Bilirubin less than or equal to 1.5 x ULN * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) less than or equal to 2.5 x ULN * Subjects who have BOTH bilirubin greater than ULN and AST/ALT greater than ULN are not eligible * Alkaline phosphatase less than or equal to 2.5 x ULN * Patients must have normal baseline thyroid function tests (thyroid-stimulating hormone \[TSH\], T3, T4); a history of hypothyroidism and/or hyperthyroidism is allowed, as long as the patient has stable well-controlled thyroid function for a minimum of 2 months * 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; 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 be capable of taking and absorbing oral medications; a patient must be clear of the following: * Any lesion, whether induced by tumor, radiation, or other conditions, which makes it difficult to swallow tablets * Prior surgical procedures affecting absorption including, but not limited to major resection of stomach or small bowel * Active peptic ulcer disease * Malabsorption syndrome * Any concomitant medications that are associated with a risk of corrected QT (QTc) prolongation and/or Torsades de Pointes should be discontinued or replaced with drugs that do not carry these risks, if possible; patients who must take medication with a risk of possible risk of Torsades de Pointes should be watched carefully for symptoms of QTc prolongation, such as syncope * Patients with personal or family history of congenital long QTc syndrome are NOT eligible * Strong inhibitors of cytochrome P-450 system (CYP)3A4 are prohibited * Strong inducers of CYP3A4 are prohibited * Concomitant use of agents with narrow therapeutic windows that are metabolized by CYP3A4, CYP2D6, or CYP2C8 is not recommended Exclusion Criteria: * Patients who have had previous treatment with pazopanib; patients who have had 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 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; 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 clinically significant cardiovascular disease; this includes: * Uncontrolled hypertension, defined as systolic greater than 140 mm Hg or diastolic greater than 90 mm Hg despite antihypertensive medications * Congenital long QT syndrome or baseline QTc greater than 480 milliseconds * 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 * Patients who have received prior treatment with an anthracycline (including doxorubicin and/or liposomal doxorubicin) must have an echocardiogram assessment and are excluded if they have an ejection fraction less than 50% * CTCAE grade 2 or greater peripheral vascular disease (at least brief \[less than 24 hours\] episodes of ischemia managed non-surgically and without permanent deficit) * History of cardiac angioplasty or stenting within 6 months prior to registration * History of coronary artery bypass graft surgery within 6 months prior to registration * Arterial thrombosis within 6 months prior to registration * 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 which are not controlled with non-enzyme inducing anticonvulsants, 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 * History of allergic reactions attributed to compounds of similar chemical or biologic composition to pazopanib * Known human immunodeficiency virus (HIV)-positive subjects on combination antiretroviral therapy * Patients with any condition that may increase the risk of gastrointestinal bleeding or gastrointestinal perforation, including: * Active peptic ulcer disease * Known gastrointestinal intraluminal metastatic lesions (gastrointestinal serosa metastatic lesions are permitted) * Inflammatory bowel disease (e.g., ulcerative colitis, Crohn's disease) * Patients with clinical symptoms or signs of gastrointestinal obstruction * Patients who require parenteral hydration and/or nutrition * Patients who are pregnant or nursing * History of hemoptysis in excess of 2.5 mL (½ teaspoon) within 8 weeks prior to first dose of pazopanib * Uncontrolled intercurrent illness including, but not limited to, psychiatric illness/social situations that would limit compliance with study requirements