Lenvatinib for the Treatment of Recurrent Hepatocellular Carcinoma After Liver Transplant

Inclusion Criteria: * Male or female * Age \>= 18 years * Eastern Cooperative Oncology Group (ECOG) performance status =\< 1 (Karnofsky \>= 60%) * Patients must have recurrent histologically or cytologically confirmed hepatocellular carcinoma that has recurred after liver transplantation and not amenable for surgical resection * Child Pugh class A * Prior orthotropic liver transplantation for curative intent * Measurable disease per Response Evaluation Criteria in Solid Tumors (RECIST) version 1.1 * Life expectancy \> 12 weeks as determined by the investigator * Hemoglobin \>= 8.0 g/dl (within 28 days of cycle 1 day 1) * Absolute neutrophil count (ANC) \>= 1,500/mcL (within 28 days of cycle 1 day 1; after at least 7 days without growth factor support or transfusion) * Platelets \>= 75,000/mcL (within 28 days of cycle 1 day 1) * International normalized ratio (INR) =\< 2.3 (within 28 days of cycle 1 day 1) * Total bilirubin =\< 3 times the institutional upper limit of normal (ULN) (within 28 days of cycle 1 day 1) * Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) =\< 5.0 times the ULN (within 28 days of cycle 1 day 1) * Albumin \>= 2.8 g/dL (within 28 days of cycle 1 day 1) * Serum creatinine =\< 1.5 x ULN or creatinine clearance \>= 60 mL/min/1.73 m\^2 for patients with creatinine levels \> 1.5 x ULN; creatinine clearance should be calculated per institutional standard (within 28 days of cycle 1 day 1) * Urinary protein =\< 1+ on dipstick or routine urinalysis or 24-hour urine demonstrating \< 1 gram of protein (within 28 days of cycle 1 day 1) * The effects of lenvatinib on the developing human fetus are unknown. For this reason females of child-bearing potential (FCBP) must have a negative serum or urine pregnancy test 72 hours prior to starting protocol therapy. Females of childbearing potential are defined as those who are not surgically sterile (i.e., bilateral salpingectomy, bilateral oophorectomy, or complete hysterectomy) or post-menopausal. Women will be considered post-menopausal if they have been amenorrheic for 12 months without an alternative medical cause. The following age-specific requirements apply: * Women \< 50 years of age would be considered post-menopausal if they have been amenorrheic for 12 months or more following cessation of exogenous hormonal treatments and if they have luteinizing hormone and follicle-stimulating hormone levels in the post-menopausal range for the institution. * Women \>= 50 years of age would be considered post-menopausal if they have been amenorrheic for 12 months or more following cessation of all exogenous hormonal treatments, had radiation-induced menopause with last menses \> 1 year ago, had chemotherapy-induced menopause with last menses \> 1 year ago * FCBP and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study treatment, and for 60 days after the last dose of protocol therapy. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study or for 60 days after the last dose of protocol therapy, she should inform the principal investigator immediately. Men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 60 days after last dose of protocol therapy. * Patients with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association (NYHA) functional classification. To be eligible for this trial, patients should be class 2B or better. * Willingness and ability of the subject to comply with scheduled visits, drug administration plan, protocol-specified laboratory tests, other study procedures, and study restrictions. * Evidence of a personally signed informed consent indicating that the subject is aware of the neoplastic nature of the disease and has been informed of the procedures to be followed, the experimental nature of the therapy, alternatives, potential risks and discomforts, potential benefits, and other pertinent aspects of study participation. Exclusion Criteria: * Prior systemic therapy with lenvatinib or another Food and Drug Administration (FDA) approved systemic therapy for hepatocellular carcinoma in the post-transplant setting. * Prior liver directed therapy is allowed, should be at least \> 28 days prior to the study enrollment. Should have at least one measurable untreated lesion by RECIST 1.1 . * Patients who have had radiotherapy within 4 weeks prior to entering the study or those who have not recovered from adverse events due to agents administered more than 4 weeks earlier (i.e., have residual toxicities \> grade 1). * Patients who are receiving any other investigational agents or an investigational device within 21 days before administration of first dose of study drugs. * History of allergic reactions attributed to compounds of similar chemical or biologic composition to lenvatinib * Patients with known brain metastases should be excluded from this clinical trial because of their poor prognosis and because they often develop progressive neurologic dysfunction that would confound the evaluation of neurologic and other adverse events. * Uncontrolled intercurrent illness including, but 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. Subjects with active hepatitis virus infection controlled with antiviral therapy are eligible. * Significant cardiovascular impairment: history of congestive heart failure greater than NYHA class II, unstable angina, myocardial infarction or stroke within 6 months of the first dose of study drug, or cardiac arrhythmia requiring medical treatment at screening. * QTc interval \> 480 ms * Uncontrolled blood pressure (systolic blood pressure \>140 mmHg or diastolic blood pressure \> 90 mmHg) in spite of an optimized regimen of antihypertensive medication. * Bleeding or thrombotic disorders or subjects at risk for severe hemorrhage. The degree of tumor invasion/infiltration of major blood vessels (e.g. carotid artery) should be considered because of the potential risk of severe hemorrhage associated with tumor shrinkage/necrosis following lenvatinib therapy. * Subjects having \> 1+ proteinuria on urine dipstick testing unless a 24-hour urine collection for quantitative assessment indicates that the urine protein is \< 1 g/24 hours. * Subjects who have not recovered adequately from any toxicity from other anti-cancer treatment regimens and/or complications from major surgery prior to starting therapy. Withhold lenvatinib for at least 1 week prior to elective surgery. Do not administer for at least 2 weeks following major surgery and until adequate wound healing. * Females who are breastfeeding or pregnant at screening or baseline (as documented by a positive beta-human chorionic gonadotropin \[beta-hCG\] or human chorionic gonadotropin \[hCG\] test with a minimum sensitivity of 25 international unites/L or equivalent units of beta-hCG \[or hCG\]). * Females of child-bearing potential must be willing to use effective contraception during study and for 60 days after the last dose. * The participant has severe hypersensitivity (\>= grade 3) to lenvatinib and/or any of its excipients. * Patients with known human immunodeficiency virus (HIV) infection * Participant with diagnosis, detection, or treatment of another type of cancer =\< 2 years prior to initiating protocol therapy (except basal or squamous cell carcinoma of the skin and cervical cancer that has been definitively treated).