Azacitidine and Abatacept in Relapsed or Refractory T-Cell Lymphoma

-INCLUSION CRITERIA 1. Participants must have a histologically or cytologically confirmed T-cell lymphoma confirmed by the Laboratory of Pathology (LP), NCI. The one of the following T-cell lymphomas are included: * Peripheral T-cell lymphoma not otherwise specified (PTCL, NOS) * Angioimmunoblastic T-cell lymphoma (AITL) * T-follicular helper (TFH) lymphoma * Anaplastic large cell lymphoma (ALCL) * Enteropathy-associated T-cell lymphoma (EATL) * Monomorphic epitheliotropic intestinal T-cell lymphoma (MEITL) * Adult T-cell leukemia/lymphoma (ATLL) * Other T-cell lymphoma (TCL) 2. Participants must have a disease that is relapsed or refractory after initial systemic treatment. 3. Participants must have evaluable disease on screening imaging or by laboratory assessment. 4. Age \>= 18 years. 5. ECOG performance status \<= 2. 6. Participants must have adequate organ and marrow function as defined below: * Absolute neutrophil count (ANC) / \>= 1,000 cells/mcL OR \>= 500 cells/mcL if bone marrow involvement with lymphoma * Platelets / \>= 50,000 cells/mcL OR \>= 25,000 cells/mcL if bone marrow involvement with lymphoma * Hemoglobin / \>= 8 g/dL (transfusions permitted) * Renal function / Serum creatinine \<= 2 mg/dL OR Glomerular filtration rate (GFR) \>= 40 mL/min/1.73 m\^2 as estimated by the Modification of Diet in Renal Disease (MDRD). Note: there is no limit if involved by lymphoma. If not on target, a 24-hour urine creatinine clearance can be used to directly measure creatinine clearance. * Total bilirubin / \<= 1.5 x upper limit of normal (ULN) * Aspartate Aminotransferase (AST) / \<= 3.0 x ULN * Alanine Aminotransferase (ALT) / \<= 3.0 x ULN * Prothrombin time (PT) / \<1.5 x ULN * Activated partial thromboplastin time (aPTT) / \<1.5 x ULN; \< 5.0 x ULN if the aPTT is prolonged because of a positive Lupus Anticoagulant International normalized ratio (INR) / \<1.5 x ULN 7. Participants, seropositive for human immunodeficiency virus (HIV), must have an undetectable HIV viral load. 8. Participants, seropositive for hepatitis C virus (HCV) infection, must have been treated and have an undetectable HCV viral load. 9. Participants who are hepatitis B core antibody (HBcAb) positive must have a hepatitis B virus (HBV) viral load result \<100 IU/mL. Note: participants who are hepatitis B surface antigen (HBsAg) positive are excluded. 10. Participants with detectable Epstein-Barr virus (EBV) or Cytomegalovirus (CMV) by polymerase chain reaction will be eligible provided they do not have end organ dysfunction from EBV or CMV infection. 11. Participants with pulmonary disease such as chronic obstructive pulmonary disease and non-infectious liver disease who otherwise meet the requirements. 12. Women of child-bearing potential (WOCBP) must agree to use effective methods of contraception (barrier, hormonal, surgical sterilization, abstinence) during the study treatment and for 6 months after the completion of the study treatment. Note: WOCBP is defined as any woman who has experienced menarche and who has not undergone successful surgical sterilization or who is not postmenopausal. Men able to father a child must agree to use an effective method of contraception (barrier, surgical sterilization, abstinence) during the study treatment and for 3 months after the completion of the study treatment. These men must not freeze or donate sperm within the same period. 13. Nursing participants must be willing to discontinue nursing from study treatment initiation through six (6) months after the last dose of the study drug(s). 14. Ability of the participant to understand and the willingness to sign a written informed consent document. Exclusion Criteria 1. Any second malignancy that requires current active systemic therapy. 2. Latent tuberculosis (TB) infection. 3. Active systemic bacterial, viral, fungal, mycobacterial, parasitic, or other infection requiring anti-infective treatment within 1 day prior to the study treatment initiation. 4. Anti-cancer therapy within 2 weeks prior to the study treatment initiation. Note: systemic steroids are allowed if \<= 100 mg/per day of prednisone (or equivalent) and were used for \<= 7 days during 2 weeks before the study treatment initiation. 5. Any investigational therapy within 2 weeks prior to the study treatment initiation. 6. Known allergy or hypersensitivity to any of the study drugs. 7. Pregnancy confirmed with beta-human chorionic gonadotropin (beta-HCG) serum or urine pregnancy test in WOCBP at screening. 8. Active central nervous system involvement with lymphoma. Previously treated central nervous system involvement with lymphoma will be allowed if \>3 months since end of treatment. 9. Uncontrolled intercurrent illness, factors, evaluated by medical history, electrocardiogram (EKG), and physical exam that would potentially increase in risk of participant.