This randomized phase II trial studies how well paclitaxel when given together with or without pazopanib hydrochloride works in treating patients with ovarian epithelial, fallopian tube, or peritoneal cavity cancer that is persistent or has come back. Drugs used in chemotherapy, such as paclitaxel work in different ways to stop the growth of tumor cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Pazopanib hydrochloride may stop the growth of tumor cells by blocking blood flow to the tumor or by blocking some of the enzymes needed for cell growth. It is not yet known whether paclitaxel is more effective when given with or without pazopanib hydrochloride in treating ovarian epithelial, fallopian tube, and peritoneal cavity cancer.

PRIMARY OBJECTIVES: I. To estimate the progression-free survival hazard ratio of the combination of weekly paclitaxel and pazopanib (pazopanib hydrochloride) compared to weekly paclitaxel and placebo in patients with persistent or recurrent ovarian, fallopian tube, or primary peritoneal cancer. SECONDARY OBJECTIVES: I. To determine the frequency and severity of adverse events as assessed by Common Terminology Criteria for Adverse Events (CTCAE). II. To estimate and compare the proportion of patients responding to therapy by Response Evaluation Criteria in Solid Tumors (RECIST), cancer antigen 125 (CA125) response, the overall survival (OS), and the duration of response in each arm. TERTIARY OBJECTIVES: I. To explore the association between plasma cytokines and angiogenic markers and progression-free and overall survival. II. To explore the association between single-nucleotide polymorphisms (SNPs) and progression-free and overall survival. OUTLINE: Patients are randomized to 1 of 2 treatment arms. ARM I: Patients receive paclitaxel intravenously (IV) over 1 hour on days 1, 8, and 15 and placebo orally (PO) daily on days 1-28. ARM II: Patients receive paclitaxel as in Arm I and pazopanib hydrochloride PO daily on days 1-28. In both arms, courses repeat every 28 days in the absence of disease progression or unacceptable toxicity. After completion of study therapy, patients are followed up every 3 months for 2 years and then every 6 months for 3 years.