Sorafenib Tosylate Following a Liver Transplant in Treating Patients With Liver Cancer

Inclusion Criteria: * Patients must have hepatocellular carcinoma (HCC) with one of the following on explant: microvascular/macrovascular invasion, tumor outside of Milan criteria, poor tumor differentiation; patients with macrovascular invasion on explant pathology will be stratified \* Additionally, the following will be included \*\* Patients with elevated surrogate markers (AFP \> 500 or PIVKA \> 400) pre transplant and with biopsy proven HCC prior to orthotopic liver transplantation (OLT) or on explant * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0-2 * Patients with a life expectancy \> 12 weeks * Patients must have completed prednisone taper within 6 weeks post OLT * Patients must be enrolled between 6 to 12 weeks post OLT * Cadaveric donors only (no living donor liver transplantation \[LDLT\] or donor after cardiac death transplantation \[DCDT\]) * No sorafenib prior to inclusion in the study * Platelet count \> 50 x 10\^9/L * Hemoglobin \>= 8.5 g/dL * Total bilirubin =\< 5 mg/dL * Alanine transaminase (ALT) and aspartate aminotransferase (AST) =\< 5 x upper limit of normal * Amylase and lipase =\< 1.5 x the upper limit of normal * Serum creatinine \< 2 x the upper limit of normal * Prothrombin time (PT) =\< 6 seconds or international normalized ratio (INR) =\< 2.3 * AFP \> 500 (pre-transplant) * PIVKA \> 400 (pre-transplant) * Patient has not received prior anti-angiogenic therapy, systemic targeted agents or systemic chemotherapy \* Prior surgical resection, chemoembolization or other local therapy prior to transplant is permitted * Women of childbearing potential must have a negative serum pregnancy test performed within 7 days prior to the start of study drug; post-menopausal women (defined as no menses for at least 1 year) and surgically sterilized women are not required to undergo a pregnancy test * Patients (men and women) of childbearing potential must agree to use adequate contraception beginning at the signing of the informed consent form (ICF) until at least 30 days after the last dose of study drug; the definition of adequate contraception will be based on the judgment of the principal investigator or a designated associate * Patient must be able to swallow and retain oral medication * Patient must exhibit the ability to understand and willingness to sign a written informed consent regarding the study and alternative treatments Exclusion Criteria: * Significant ongoing immunologic rejection based on pathology and clinical diagnosis (from time of transplant until randomization) * Use of T cell depleting agents for prevention or treatment of rejection at any point prior to or after enrollment in the study * Patient with documented evidence of metastatic disease * 100% tumor necrosis on explant pathology * Use of mammalian target of rapamycin (mTOR) inhibitors prior to transplant and as post-transplant immunosuppression * Use of alemtuzumab * Living donor liver transplant (LDLT) or donation after cardiac death transplant (DCDT) * Human immunodeficiency virus (HIV) positive patients * Hepatitis C virus (HCV) recurrence at the time of randomization * Use of direct acting antivirals for HCV recurrence * Requirement of re-transplantation for primary non function * Uncontrolled hypertension, defined as systolic blood pressure \> 140 mmHg or diastolic pressure \> 90 mmHg, despite optimal medical management * Active or clinically significant cardiac disease including: * Congestive heart failure - New York Heart Association (NYHA) \> class II * Coronary artery disease * Cardiac arrhythmias requiring anti-arrhythmic therapy other than beta blockers or digoxin * Unstable angina or new-onset angina within 3 months before randomization, or myocardial infarction (MI) within 6 months before randomization * Clinically active serious infection documented by positive cultures or an incomplete course of treatment for bacteremia or fungemia * Evidence or history of bleeding diathesis or coagulopathy * Patients with any pulmonary hemorrhage/bleeding event grade 2 or higher within 4 weeks before randomization * Patients with thrombotic, embolic, venous, or arterial events, such as cerebrovascular accident (including transient ischemic attacks \[TIAs\]) within 6 months before randomization * Prior use of raf-kinase inhibitors (sorafenib), vascular endothelial growth factor (VEGF) inhibitors, mitogen-activated protein kinase (MAPK)/extracellular-signal-related kinase (ERK) kinase (MEK) inhibitors, or farnesyl transferase inhibitors * Patients using cytochrome P450 3A4 (CYP3A4) inducers (eg, phenytoin, carbamazepine, phenobarbital, St. John's Wort dexamethasone) at a dose of greater than 16 mg daily, or rifampin and/or rifabutin within 28 days before randomization * Patients with non-HCC malignancy except those with DCIS (ductal carcinoma in situ), cervical cancer in-situ, basal cell or superficial bladder tumor; patients surviving a cancer that was curatively treated and without evidence of recurrence for more than 3 years before randomization are allowed * Presence of a non-healing wound, non-healing ulcer, or bone fracture * Known or suspected allergy or hypersensitivity to any of the study drugs, study drug classes, or excipients of the formulations given during the course of this trial * Any malabsorption condition * Women who are pregnant or breast-feeding * Inability to comply with the protocol and/or not willing or not available for follow-up assessments * Patients with fibrolamellar HCC, cholangiocarcinoma, and combined HCC-cholangiocarcinoma * Any condition which, in the investigator's opinion, makes the patient unsuitable for trial participation * Prior use of any systemic chemotherapy for HCC * Prior use of systemic investigational agents for HCC * Prior use of raf-kinase inhibitors (sorafenib), VEGF inhibitors, MEK inhibitors or farnesyl transferase inhibitors * Use of biologic response modifiers, such as granulocyte colony-stimulating factor (G-CSF), will not be permitted within 3 weeks prior to study entry; G-CSF and other hematopoietic growth factors may be used in the management of acute toxicity, such as febrile neutropenia, when clinically indicated or at the discretion of the investigator; however, they may not be administered to prevent a dose reduction; patients taking chronic erythropoietin are permitted, provided no dose adjustment is undertaken within 1 month prior to randomization or during the study * Autologous bone marrow transplant or stem cell rescue within four months of start of study drug * Concomitant treatment with rifampin and St. John's wort * Concomitant oral mTOR inhibitor treatment * Use of direct acting antivirals for HCV recurrence * Use of T-cell depleting agents * Use of alemtuzumab * Anticoagulation, as described below, is allowed: * Vitamin-K antagonists (e.g., warfarin) \*\* Low dose warfarin (1 mg orally, once daily) with PT-INR =\< 1.5 x ULN is permitted; patients taking concomitant warfarin should be monitored regularly for changes in PT, PT-INR or clinical bleeding episodes * Low dose aspirin (=\< 100 mg daily). * Heparins and heparinoids Use of any other investigational drug