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.

PRIMARY OBJECTIVES:

I. To determine the recommended phase II dose (RPTD) of combination lenvatinib mesylate (lenvatinib) and weekly paclitaxel in patients with recurrent endometrial, epithelial ovarian, primary peritoneal, or fallopian tube carcinoma.

SECONDARY OBJECTIVES:

I. To determine the safety and tolerability of combination lenvatinib and weekly paclitaxel in patients with recurrent endometrial, epithelial ovarian, primary peritoneal, or fallopian tube carcinoma.

II. To explore the objective antitumor activity (complete and partial response) of combination lenvatinib and weekly paclitaxel as measured by Response Evaluation Criteria in Solid Tumors (RECIST) criteria.

III. To measure the progression free survival. IV. To evaluate the pharmacokinetics of combination paclitaxel and lenvatinib.

OUTLINE: This is a dose -escalation study of lenvatinib mesylate.

Patients receive paclitaxel intravenously (IV) over 1 hour on days 1, 8, and 15 and lenvatinib mesylate orally (PO) once daily (QD) on days 1-28. Courses repeat every 28 days in the absence of disease progression or unacceptable toxicity.

After completion of study treatment, patients are followed up every 3 months for 2 years and then every 3 months for 3 years.

Inclusion Criteria:

• Women with histologically confirmed endometrial cancer, epithelial ovarian cancer, fallopian tube cancer, or primary peritoneal cancer (all histological subtypes)who have disease progression after treatment with available therapies that are known to confer clinical benefit or who are intolerant to prior treatment
• Patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as ≥ 20 mm with conventional techniques or as ≥10 mm with spiral computerized tomography (CT) scan, magnetic resonance imaging (MRI), or calipers by clinical exam
• Patients must have received prior treatment with a platinum containing regimen and may have received an unlimited number of prior regimens (including prior taxanes)
• Patients with ovarian, Fallopian tube or primary peritoneal cancer must be platinum resistant (progression < 6 months after completion of a platinum containing regimen)
• Patients may have received prior targeted therapy such as bevacizumab
• Eastern Cooperative Oncology Group performance status =< 1
• Leukocytes >= 3,000/mcL (microliter)
• Absolute neutrophil count >= 1,500/mcL
• Platelets >= 100,000/mcL
• Hemoglobin >=8.0 g/dL
• Total bilirubin within normal institutional limits
• Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase [SGOT])/alanine aminotransferase (ALT) (serum glutamic pyruvic transaminase [SGPT]) =< 2.5 × institutional upper limit of normal
• Creatinine < 1.5 mg/dL X ULN OR creatinine clearance >= 30 mL/min for patients with creatinine levels above institutional normal
• Urine protein by dipstick <1+ or UPC =< 1.0 by urinalysis
• Patients with chronic hypertension that is well controlled with systolic blood pressure of < 140 mmHg or diastolic blood pressure of < 90 mmHg, and in whom there has been no change in blood pressure medication in the last two weeks, are eligible
• Patients who have a history of deep vein thrombosis (DVT) or pulmonary embolus and are stable on anticoagulation for > 1 month are eligible