Tivozanib in Treating Patients With Liver Cancer That Is Metastatic or Cannot Be Removed by Surgery

Inclusion Criteria: * Advanced staged HCC (unresectable and not amenable to local or regional therapy; or metastatic HCC); the diagnosis of HCC should be based on at least one of the following: * Magnetic resonance imaging (MRI) or computed tomography (CT) consistent with liver cirrhosis AND at least one solid liver lesion measuring \>= 2 cm, with characteristics arterial enhancement and venous washout regardless of alpha-fetoprotein (AFP) levels * AFP \>= 400 ng/mL AND evidence of at least one solid liver lesion \>= 2 cm regardless of specific imaging characteristics on CT or MRI * Histological/cytology biopsy confirming HCC * Patients must have measurable disease per RECIST 1.1 criteria defined as at least one lesion that can be accurately measured in at least one dimension, and that has not been the target of local or regional therapy including transarterial chemoembolization, intra-arterial chemotherapy, ethanol or radiofrequency ablation * Life expectancy of greater than 3 months * Child-Pugh liver function class A * Aspartate aminotransferase (AST) =\< 5 x institutional upper limits of normal (ULN) * Total bilirubin =\< 3 mg/dL * International normalized ratio (INR) =\< 2.0 (unless due to therapeutic warfarin use) * Serum albumin \> 2.8 g/dL * Creatinine =\< 1.5 x institutional ULN * Absolute neutrophil count (ANC) \>= 1200/mm\^3 * Platelets \>= 60,000/mm\^3 * Hemoglobin (Hgb) \>= 8.5 g/dL * Patients must not have any evidence of bleeding diathesis or active gas…