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 c…