Re-Induction Therapy for Relapsed Pediatric T-Cell Acute Lymphoblastic Leukemia or Lymphoma

Inclusion Criteria: * Participants must have relapsed or refractory acute lymphoblastic leukemia or lymphoma (ALL): * Stratum I: T-cell lymphoblastic leukemia or lymphoma in first relapse or refractory to one or two courses of frontline induction therapy. * Stratum II: B-cell or T-cell lymphoblastic leukemia or lymphoma in second or third relapse or refractory to 2 or 3 induction or re-induction attempts. Patients with Ph+ ALL must be refractory or relapsed after treatment with regimen that included a tyrosine kinase inhibitor (TKI). * Relapse in ALL is defined as the reappearance (in a patient who has previously achieved remission) of leukemic blasts in the bone marrow. * Should flow cytometric analyses suggest relapse (by the reappearance of a similar immunophenotype to the original leukemia) in the presence of \<5% blasts morphologically, a repeat bone marrow test is recommended to confirm relapse. * Molecular or genetic relapse is characterized by the reappearance of a cytogenetic or molecular abnormality. * Age is ≤ 21 years (participant has not yet reached 22nd birthday). * Able to swallow capsules. * Karnofsky or Lansky performance score is ≥ 60%. The Lansky performance score should be used for participants \< 16 years and the Karnofsky performance score for participants ≥ 16 years. * Prior therapy: * There is no waiting period for participants who relapse while receiving therapy if they are free from side effects attributable to such therapy. * Emergent radiation therapy, one dose of intrathecal chemotherapy and up to 7 days of steroids or hydroxyurea are permitted before start of treatment in participants who relapse after completion of frontline therapy. Other circumstances must be cleared by PI or medical designee. * At least 90 days have elapsed since bone marrow transplant and participant is off immune suppression for ≥ 2 weeks, if applicable. * Adequate renal function defined as glomerular filtration rate ≥ 60 cc/min/1.73m\^2 or serum creatinine based on age as follows: * If age is 1 to 2 years, then maximum serum creatinine (mg/dL) is 0.6 for males or females. * If age is 2 to 6 years, then maximum serum creatinine (mg/dL) is 0.8 for males or females. * If age is 6 to 10 years, then maximum serum creatinine (mg/dL) is 1 for males or females. * If age is 10 to \<13 years, then maximum serum creatinine (mg/dL) is 1.2 for males or females. * If age is 13 to 16 years, then maximum serum creatinine (mg/dl) is 1.5 for males or 1.4 for females. * If age is \> 16 years, then maximum serum creatinine (mg/dl) is 1.7 for males or 1.4 for females. * Adequate hepatic function defined as: * Direct bilirubin ≤ 1.4 mg/dL (if total bilirubin \> 1.4 mg/dL) AND * AST and ALT \< 5 x ULN for age. * Adequate cardiac function defined as shortening fraction of ≥ 27% or ejection fraction ≥ 45%. * Lymphoma participants without bone marrow involvement must have: * Absolute neutrophil count (ANC) \>1,000/mm3, AND * Platelet count ≥50,000/mm\^3 (without transfusion support) * NOTE: These criteria are waived for participants with leukemia or lymphoma participants with bone marrow involvement. * Written, informed consent and assent following Institutional Review Board, NCI, FDA and OHRP guidelines. Exclusion Criteria: * Prior histone deacytylases (HDAC), DAC, HSP90 inhibitors or valproic acid for treatment of cancer. * Patients who will need valproic acid for any medical condition during the study or within 5 days prior to first panobinostat treatment. * Impaired cardiac function or clinically significant cardiac diseases, history of arrhythmia (including ventricular fibrillation or torsade de pointes), bradycardia \<50 bpm, screening ECG with prolonged QTc or uncontrolled hypertension. * Impairment of gastrointestinal (GI) function or GI disease that may significantly alter the absorption of panobinostat. * Patients with diarrhea \> CTCAE grade 2. * Other concurrent severe and/or uncontrolled medical conditions (e.g., uncontrolled diabetes or active or uncontrolled infection) including abnormal laboratory values, that could cause unacceptable safety risks or compromise compliance with the protocol. * Patients using medications that have a relative risk of prolonging the QT interval or inducing torsade de pointes if treatment cannot be discontinued or switched to a different medication prior to starting treatment. * Patients who have received targeted agents within 2 weeks or within 5 half-lives of the agent and active metabolites (whichever is longer) and who have not recovered from side effects of those therapies. * Patients who have undergone major surgery ≤ 4 weeks prior to starting treatment or who have not recovered from side effects of such therapy. * Patients with known positivity for human immunodeficiency virus (HIV) or hepatitis B/C. * Inability to swallow capsules. * Active, uncontrolled infection or severe concurrent medical disease, including but not limited to congestive heart failure, cardiac arrhythmias, or psychiatric illness. * Isolated extramedullary relapse (leukemia) or isolated CNS lymphoma. * Pregnant or lactating (female participant of childbearing potential must have negative serum or urine pregnancy test required within 7 days prior to start of treatment). Male or female of reproductive potential has agreed to use effective contraception method for duration of study treatment. * Down syndrome. * Inability or unwillingness or research participant or legal guardian/representative to give written informed consent.