A Phase Ib Study of Belinostat With RDHAP Chemotherapy (Dexamethasone, Cytarabine, Cisplatinum) in Adults With Relapsed or Refractory Diffuse Large B-cell Lymphoma

Study Groups: If participant is found to be eligible to take part in this study, they will be assigned to a dose level of belinostat based on when they join this study. Up to 4 dose levels of belinostat will be tested. Between 3-6 participants will be enrolled at each dose level. The first group of participants will receive the lowest dose level. Each new group will receive a higher dose than the group before it, if no intolerable side effects were seen. This will continue until the highest tolerable dose of belinostat is found. All participants will receive the same dose level of the standard chemotherapy drugs. Study Drug Administration: Participant will receive belinostat with standard chemotherapy for up to two 21-day cycles. Participant will receive belinostat by vein on Days 1 and 2 of each cycle. The infusions will last 24 hours. Participant will receive rituximab and cisplatin by vein on Day 2 of each cycle. Participant will receive cytarabine by vein on Day 2 or 3 of each cycle, depending on when they join this study. Rituximab will be given over 4-5 hours, cisplatin will be given over 3-4 hours, and cytarabine will be given over 1 hour. Participant will take dexamethasone once a day by mouth on Days 2-5 of both study cycles. Blood (about 5 tablespoons) and bone marrow will be collected about 10 business days (Monday-Friday) before participant begins receiving belinostat. The blood and bone marrow will be used for peripheral blood mononuclear cell (PBMC) testing. This testing is designed to learn if participant will respond to treatment. To collect a bone marrow biopsy, an area of the hip or other site is numbed with anesthetic, and a small amount of bone marrow is withdrawn through a large needle. Participant will also receive filgrastim, pegfilgrastim, SMX, valacyclovir, ciprofloxacin and fluconazole to reduce side effects. Participant's study doctor will discuss with them when and how they will take these drugs. If participant's doctor thinks it is necessary, they will also have a bone marrow biopsy and aspirate performed to check the status of the disease within 4 weeks before they begin receiving belinostat. To collect a bone marrow biopsy/aspirate, an area of the hip or other site is numbed with anesthetic, and a small amount of bone marrow and bone is withdrawn through a large needle. Study Visits: Within 5 working/business days before the start of each cycle: * Participant will have a physical exam. * Blood (about 4-6 tablespoons) will be drawn for routine tests. One (1) time a week after the completion of therapy during each cycle, blood (about 4-6 tablespoons) will be drawn for routine tests. Blood (about 4 tablespoons) will be drawn on Days 1,2, 3, and 21 (+/- 72 hours) of Cycles 1 and 2 for PBMC testing. Length of Treatment: Participant may continue taking belinostat combined with chemotherapy for up to 3 cycles. Participant will no longer be able to take the study drugs if the disease gets worse, if intolerable side effects occur, or if they are unable to follow study directions. Patient's participation on the study will be over after the follow-up visits. End-of-Dosing Visit: Within 3 weeks after the start of the last cycle: * Participant will have a physical exam. * Blood (about 4-6 tablespoons) will be drawn for routine and PBMC testing. * Participant will have an FDG-PET/CT scan to check the status of the disease. * If participant's doctor thinks it is necessary, they may have a bone marrow biopsy and aspirate to check the status of the disease * If participant's doctor thinks it is necessary, they may have a core needle biopsy to check the status of the disease. To perform a core biopsy, a sample of tissue is removed using a hollow core needle that has a cutting edge. This is an investigational study. Belinostat is FDA approved to treat T-cell lymphoma. Its use in this study is investigational. The standard chemotherapy drugs are commercially available and FDA approved to treat DLBCL. Up to 40 participants will be enrolled on this study. All will be enrolled at MD Anderson.