This phase I trial studies the side effects and best dose of lenvatinib mesylate when given together with paclitaxel in treating patients with endometrial, ovarian, fallopian tube, or primary peritoneal cancer that has come back or grown. Lenvatinib mesylate may stop the growth of tumor cells by blocking a protein needed for cell growth and may block the growth of new blood vessels necessary for tumor growth. 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. Giving lenvatinib mesylate and paclitaxel together may work better in treating patients with endometrial, ovarian, fallopian tube, or primary peritoneal cancer.