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 gastrointestinal bleeding * Patients must not be known to be human immunodeficiency virus (HIV) positive * Patients must not have other uncontrolled intercurrent illnesses (excluding HBV or HCV); this includes (but is not limited to) ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements * Sexually active fertile patients (male and female), and their partners, must agree to use medically accepted methods of contraception during the course of the study and for 3 months after the last dose of the study drug * Female patients of childbearing potential must have a negative pregnancy test at screening * Have an Eastern Cooperative Oncology Group (ECOG) performance status of =\< 2 * Subject 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: * Patients who have had prior anti-angiogenic therapy, including but not limited to sorafenib, brivanib, bevacizumab, or sunitinib * Patients who have had any prior line of systemic therapy including cytotoxic agents or molecularly targeted agents for advanced/unresectable disease; any number of prior regional therapies with transarterial chemoembolization (TACE), brachytherapy with yttrium-90 microsphere, intra-arterial chemotherapy, surgery, or ablative therapy are allowed * Prior liver transplantation and on immunosuppression * Known symptomatic or uncontrolled brain metastases or epidural disease * Patient has a corrected QT interval (QTcF) \> 500 ms at screening * The patient is unable to swallow pills or diagnosed with a gastrointestinal disorder that are likely to interfere with the absorption of the study drug or with the patient's ability to take regular oral medication * The patient is pregnant or breastfeeding * Patients with second primary cancer (except adequately treated nonmelanoma skin cancer, curatively treated in-situ carcinoma of the cervix or superficial bladder cancer, or other solid tumors including lymphoma without bone marrow involvement curatively treated with no evidence of disease for \>= 5 years) * The patient has a previously-identified allergy or hypersensitivity to components of the study treatment formulation * Patients receiving any medications or substances that are strong inducers of cytochrome P450, family 3, subfamily A, polypeptide 4 (CYP3A4) are ineligible; moderate inducers of CYP3A4 should be used with caution * Urine protein: creatinine ratio \> 1