Sorafenib and Vorinostat in Treating Patients With Advanced Liver Cancer

Inclusion Criteria: * Diagnosis of HCC by biopsy-proven pathologic diagnosis or by clinical criteria as defined below: \* Clinical criteria to be met if patient has a history of cirrhosis or chronic hepatitis B infection: * Imaging abnormalities \> 1 cm in size with classic enhancement by magnetic resonance imaging (MRI) or triple-phase computed tomography (CT) scan * Alpha-fetoprotein (AFP) of any value * Performance status Eastern Cooperative Oncology Group (ECOG) =\< 1 * If cirrhosis, Child-Pugh classification A or B * Total bilirubin =\< 3.0 mg/dL * Creatinine =\< 1.5 x upper limit of normal for the laboratory * International normalized ratio (INR) =\< 1.7 (if not due to anticoagulants) * Absolute neutrophil count (ANC) \>= 1,500/mm\^3 * Platelets \>= 80,000/mm\^3 * Hemoglobin (Hgb) \>= 8.5 g/dL (transfusion or erythropoietin-like substances not permitted prior to baseline evaluation) * Any prior therapies such as surgery, chemoembolization, radiofrequency ablation, and alcohol injection are allowed as long as toxicity from such prior therapy is =\< grade 1 * Prior sorafenib is allowed as long as toxicity from ongoing is ≤ grade 2 and prior intolerance of 400 mg sorafenib PO daily is felt amenable, by the principal investigator, to supportive care measures or dose modifications. * Measurable or evaluable disease by Response Evaluation Criteria in Solid Tumors (RECIST) criteria version (v) 1.1 mRECIST or elevated AFP * Ability to understand and willingness to sign a written informed consent; a signed informed consent must be obtained prior to any study specific procedures * Women of childbearing potential must have a negative pregnancy test performed within 2 weeks prior to the start of treatment * Women of childbearing potential and men must agree to use a medically accepted form of birth control for the duration of study participation and for 4 months following completion of study treatment Exclusion Criteria: * Candidate for curative therapy including surgical resection or orthotopic liver transplantation * Known central nervous system metastasis * Any investigational agent within 4 weeks of first dose of study treatment * Known intolerance of vorinostat * Unable to swallow medication * Unable to swallow medication; suspected malabsorption * Active alcohol abuse * Contraindication to antiangiogenic agents, including: * Pulmonary hemorrhage/bleeding event \>= grade 2 within 4 weeks of first dose of study drug * Any other hemorrhage/bleeding event \>= grade 3 within 4 weeks of first dose of study treatment * Serious non-healing wound, ulcer, or bone fracture * Thrombolic or embolic events such as a cerebrovascular accident including transient ischemic attacks within the past 6 months; hepatic portal vein thrombus is not considered an exclusion criterion * Major cardiac dysfunction, such as uncontrolled angina, congestive heart failure with New York Heart Association (NYHA) class III or higher, known left ventricular ejection fraction less than 40% * Systolic blood pressure \> 160 mmHg or diastolic pressure \> 90 mmHg despite optimal medical management * Significant lung disease (oxygen \[O2\] saturation less than 88% in room air) * Serious uncontrolled infection; known human immunodeficiency virus (HIV)-seropositivity requiring retroviral therapy, or diagnosis of acquired immune deficiency syndrome (AIDS); diagnosis of chronic hepatitis B or C allowed * Medical, psychological, or social conditions that, in the opinion of the investigator, may increase the patient's risk or interfere with the patient's participation in the study or hinder evaluation of the study results