Lenvatinib and Weekly Paclitaxel for Patients With Recurrent Endometrial or Ovarian Cancer

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