This randomized phase II trial studies how well giving cabozantinib-s-malate or paclitaxel works in treating patients with persistent or recurrent epithelial ovarian, fallopian tube, or primary peritoneal cavity cancer. Cabozantinib-s-malate may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth. Drugs used in chemotherapy, such as paclitaxel, work in different ways to stop the growth of tumor cells, either by killing the cells or by stopping them from dividing. It is not yet known whether cabozantinib-s-malate or paclitaxel is more effective at treating patients with persistent or recurrent epithelial ovarian, fallopian tube, or primary peritoneal cavity cancer.