Abemaciclib and Nivolumab for Subjects With Hepatocellular Carcinoma

Inclusion Criteria: 1. Subjects must have hepatocellular carcinoma (HCC) that is inoperable (where surgery is not indicated due to disease extent, co-morbidities, or other technical reasons) 2. Histologic confirmation of HCC is not required for screening but is required prior to initiation of study treatment. Subjects with hepatocholangiocarcinoma or cholangiocarcinoma are not eligible. 3. Tumor must be positive for retinoblastoma (RB) expression by immunohistochemistry 4. Age \> 18 years and ability to understand and the willingness to sign a written informed consent document. 5. ECOG performance status of 0 or 1 6. Childs-Pugh score of \<7 7. Life expectancy of at least 12 weeks 8. Must be able to swallow tablets 9. Must be willing to comply with protocol procedures (including completion of diaries and outcome measures) 10. Local or loco-regional therapy to the liver (i.e. surgery, radiation therapy, hepatic arterial embolization, chemoembolization, radiofrequency ablation, percutaneous ethanol injection, or cryoablation) must have been completed \> 4 weeks prior to enrollment 11. Must be willing to undergo a pretreatment and on-treatment biopsy and have a tumor site that is accessible for core needle biopsy 12. Measurable or evaluable disease as defined by RECIST v. 1.1 13. Women of childbearing potential must have a negative serum pregnancy test performed within 7 days of the first dose of abemaciclib (see Appendix A for definition of childbearing potential). Female subjects of childbearing potential must use an approved contraceptive method (detailed in Appendix A) for the duration of the study and an additional 3 weeks after the final dose of abemaciclib. 14. Subjects with hepatitis B must have an HBV viral load \< 100 IU/mL by PCR during screening 15. Must have adequate organ and hematopoietic function as defined below: Exclusion Criteria: 1. Any history of a serious medical or psychiatric condition that would prevent the subject from signing the informed consent form 2. Pregnant or breastfeeding 3. Use of any chemotherapy within 3 weeks prior to the first study treatment date 4. Use of any experimental therapy within 4 weeks or 5 half-lives, whichever is longer, prior to the first study treatment date 5. Use of radiation within 2 weeks prior to the first study treatment date (4 weeks if radiation to liver as per section 4.1) 6. Prior treatment with a CDK 4/6 inhibitor 7. Prior treatment with a PD-1 or PD-L1 inhibitor 8. Those who have not recovered from adverse events \< Grade 1 from prior therapy, with the exceptions of alopecia of any grade or stable peripheral neuropathy \< Grade 2 9. Subjects may not receive concomitant anticancer agents or radiation. Antiviral agents aimed at treating infectious hepatitis are permitted 10. History of or suspected hypersensitivity to nivolumab or abemaciclib 11. Uncontrolled ascites 12. Esophageal varices requiring treatment within the past 6 months (banding or medication) 13. Subjects with uncontrolled brain metastases. Subjects with brain metastases must have stable neurological status following local therapy (surgery or radiation) for at least 4 weeks prior to first study treatment and must be off of steroids related to the brain metastases. 14. Any concurrent condition requiring the continued or anticipated use of systemic steroids beyond physiologic replacement dosing (excluding non-systemic inhaled, topical skin, nasal, and/or ophthalmic corticosteroids). All other systemic corticosteroids above physiologic replacement dosing must be discontinued at least 4 weeks prior to first study treatment 15. Active drug or alcohol use or dependence as documented in the chart that, in the opinion of the investigator, would interfere with adherence to study requirements 16. Active bacterial or fungal infection requiring IV therapy at the start of protocol treatment 17. A second primary malignancy that, in the judgment of the investigator, may affect the interpretation of results 18. Any illness or condition that in the opinion of the investigator may affect the safety of the subject or the evaluation of any study endpoint (e.g. interstitial lung disease, severe dyspnea at rest or requiring oxygen therapy, history of major surgical resection involving the stomach or small bowel, or preexisting Crohn's disease or ulcerative colitis or a preexisting chronic condition resulting in baseline Grade 2 or higher diarrhea) 19. Personal history of ventricular tachycardia, ventricular fibrillation, or sudden cardiac arrest 20. Prior organ allograft or allogeneic bone marrow transplantation 21. Pre-existing thyroid abnormality with thyroid function that cannot be maintained in the normal range with medication 22. Active autoimmune disease, except for vitiligo, type 1 diabetes mellitus, asthma, atopic dermatitis, or endocrinopathies manageable by hormone replacement; other autoimmune conditions may be allowable at the discretion of the Principal Investigator 23. Any other conditions judged by the investigator that would limit the evaluation of the subject 24. HIV positive by PCR