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.