Tivozanib Before Surgery in Treating Patients With Localized Kidney Cancer

Inclusion Criteria: * Biopsy proven non-metastatic clear cell RCC, as per radiographic studies is T2 - T3a based on the American Joint Committee on Cancer (AJCC) 6th edition criteria * Have an Eastern Cooperative Oncology Group (ECOG) performance status of =\< 2 * Hemoglobin \>= 10 gm/dL * Absolute neutrophil count (ANC) \>= 1.5 X 10\^9/L * Platelets \>= 100 X 10\^9/L * Total bilirubin \< 1.5 X upper limit of normal (ULN) * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) \< 2.5 X ULN * International normalization ratio (INR) \< 1.5 * Activated partial thromboplastin time (aPTT) \< 1.2 X ULN * Serum creatinine \< 1.5 mg/dL or if \>= 1.5 mg/dL: calculated creatinine clearance (CrCL) \> 30 mL/min based on Cockroft-Gault formula * Must have the ability to swallow and retain oral medication * Patients of childbearing potential must agree to use acceptable contraceptive methods (e.g., double barrier) during treatment * Patient or legal representative must understand the investigational nature of this study and sign an Independent Ethics Committee/Institutional Review Board approved written informed consent form prior to receiving any study related procedure Exclusion Criteria: * Pregnant or nursing female patients * Unwilling or unable to follow protocol requirements * Any condition which in the Investigator's opinion deems the patient an unsuitable candidate to receive tivozanib (solitary kidney, multiple tumors in the same or contralateral kidney) * Not a candidate for surgery * Received an investigational agent within 30 days prior to enrollment * Non-clear cell or sarcomatoid histology * Patients with metastatic disease at presentation * Prior therapy with tyrosine kinase inhibitor for RCC * A second primary malignancy (except squamous and basal cell carcinoma of skin) in the past 3 years * Active or chronic infections * Significant cardiovascular disease, including: * Clinically symptomatic left ventricular failure * Uncontrolled hypertension: systolic blood pressure of \> 150 mmHg or diastolic blood pressure of \> 100 mmHg documented on 2 consecutive measurements taken at least 24 hours apart * Myocardial infarction, severe angina, or unstable angina within 6 months prior to administration of first dose of tivozanib * History of serious ventricular arrhythmia (i.e., ventricular tachycardia or ventricular fibrillation) * Cardiac arrhythmias requiring anti-arrhythmic medications (except for atrial fibrillation that is well controlled with anti-arrhythmic medication) * Coronary or peripheral artery bypass graft within 6 months of screening * History of coronary artery disease or peripheral arterial disease * History of stroke or carotid endarterectomy * Patients who are taking cytochrome P450 system (CYP)3A4 inducers are excluded; patients taking CYP3A4 inducers that can be safely replaced with another agent may be enrolled after a 5 day washout period