Vorinostat and Combination Chemotherapy Before Donor Stem Cell Transplantation for the Treatment of Relapsed Aggressive B-cell or T-cell Non-Hodgkin Lymphoma

Inclusion Criteria:

* Relapsed aggressive B-cell non-Hodgkin lymphoma (B-NHL) (diffuse large B-cell lymphoma \[DLBCL\], transformed B-NHL, high-grade B-cell lymphoma \[HGBL\] or Burkitt) or T-cell non-Hodgkin lymphoma (T-NHL) who meet both of the following criteria: a. High or very high disease risk index (DRI), and b. No response to at least 1 line of salvage chemotherapy, or relapse after a prior autologous SCT
* An 8/8 human leukocyte antigen (HLA) matched (high resolution typing at A, B, C, DRB1) sibling or unrelated donor, or a haploidentical donor
* Left ventricular ejection fraction (EF) \>= 45%
* Forced expiratory volume in 1 second (FEV1), forced vital capacity (FVC) and corrected diffusion capacity of the lung for carbon monoxide (DLCO) \>= 50%
* Estimated serum creatinine clearance \>= 50 ml/min (using the Cockcroft-Gault formula)
* Serum bilirubin =\< 2 x upper limit of normal
* Serum glutamate pyruvate transaminase (SGPT) =\< 2 x upper limit of normal
* Able to sign informed consent
* Men and women of reproductive potential must agree to follow accepted birth control methods for the duration of the study. Female subject is either post-menopausal or surgically sterilized or willing to use an acceptable method of birth control (i.e., a hormonal contraceptive, intra-uterine device, diaphragm with spermicide, condom with spermicide, or abstinence) for the duration of the study. Male subject agrees to use an acceptable method for contraception for the duration of the study

Exclusion Criteria:

* Patient with active central nervous system (CNS) disease
* Pregnancy (positive beta human chorionic gonadotropin \[HCG\] test in a woman with child bearing potential defined as not post-menopausal for 12 months or no previous surgical sterilization) or currently breast-feeding. Pregnancy testing is not required for post-menopausal or surgically sterilized women
* Active hepatitis B, either active carrier (hepatitis B surface antigen positive \[HBsAg +\]) or viremic (hepatitis B virus \[HBV\] DNA \>= 10,000 copies/mL, or \>= 2,000 IU/mL)
* Evidence of either cirrhosis or stage 3-4 liver fibrosis in patients with chronic hepatitis C or positive hepatitis C serology
* Human immunodeficiency virus (HIV) infection
* Active uncontrolled bacterial, viral or fungal infections
* Exposure to other investigational drugs within 4 weeks before enrollment
* Grade \>= 3 non-hematologic toxicity from previous therapy that has not resolved to =\< grade 1
* Radiation therapy to head and neck (excluding eyes), and internal organs of chest, abdomen or pelvis in the month prior to enrollment
* Prior whole brain irradiation
* Prior autologous SCT in the prior 3 months