HDAC Inhibitor AR-42 and Pomalidomide in Treating Patients With Relapsed Multiple Myeloma

Inclusion Criteria: * Patients with measurable disease as defined by any of the following: * Serum M-protein \>= 0.5 g/dl (\>= 500 mg/dL) * Urine monoclonal protein \>= 200 mg/24h * Involved free light chain (FLC) level \>= 10 mg/dl (\>= 100 mg/l) and an abnormal serum free light chain ratio (\< 0.26, or \> 1.65) * Patients must have previously received Lenalidomide and proteasome inhibitor. * Patients must be lenalidomide failures: disease progression on a prior lenalidomide-based therapy or progression within 60 days of the last dose of a lenalidomide; patients should have received at least 2 cycles of a lenalidomide-based regimen at standard doses to be evaluable for refractoriness; prior intolerance to lenalidomide does not exclude participation in the study except in cases of severe allergic reaction * Prior radiation is permitted; however, at least 2 weeks must have elapsed since the completion of therapy and patients must have recovered from all therapy-associated toxicities to no greater than grade 1 at the time of registration; patients with symptomatic disease may receive palliative corticosteroids up to 1 week before initiating therapy * Patient must have received 2 or more prior lines of systemic therapy for myeloma; patients must be off last treatment for at least 2 weeks (wks) by the beginning of treatment on this protocol * Patients must have a Karnofsky performance score of 50% or greater * Patients must have absolute neutrophil count (ANC) \> 1000/uL * Platelets \>= 75,000/uL * Total bilirubin =\< 1.5 mg/dL * Alkaline phosphatase =\< 4 x institutional upper limit of normal (IULN) * Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) =\< 3 x IULN * Patients must have a serum creatinine limit of ≤1.5 ULN or creatinine clearance of ≥60 ml/min measured within 14 days of registration. * All study participants must be registered into the mandatory POMALYST (pomalidomide) Risk Evaluation and Mitigation Strategy (REMS) program, and be willing and able to comply with the requirements of the POMALYST REMS program * Females of childbearing potential (FCBP) must have a negative serum or urine pregnancy test with a sensitivity of at least 50 mIU/mL prior to starting therapy and prior to beginning another cycle (if applicable); women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation; should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately; a female of childbearing potential (FCBP) is a sexually mature woman who: 1) has not undergone a hysterectomy or bilateral oophorectomy; or 2) has not been naturally postmenopausal for at least 24 consecutive months (i.e., has had menses at any time in the preceding 24 consecutive months) * The patient must be willing to comply with fertility requirements as below: * Male patients must agree to use an adequate method of contraception for the duration of the study and for 28 days afterwards * Female patients must be either postmenopausal, free from menses \>= 2 yrs, surgically sterilized, willing to use two adequate barrier methods of contraception to prevent pregnancy, or agree to abstain from heterosexual activity starting with screening (4 weeks prior to initiating treatment) and 28 days after treatment, per POMALYST REMS™ program * Patients must agree not to donate blood, sperm/ova during the course of taking protocol therapy and for at least 4 weeks after stopping treatment Exclusion Criteria: * History of severe allergic reaction, including erythema nodosum, to lenalidomide * Patients unable to receive adequate thromboprophylaxis in combination with pomalidomide * Patients who have received investigational 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 with a mean QT interval corrected by Bazett's formula (QTcB) \> 450 msec in males and \> 470 msec in females * Patients with known positivity for human immunodeficiency virus (HIV) or hepatitis C * Patients with active (untreated or relapsed) central nervous system (CNS) metastasis of the patient's myeloma * Any other medical condition, including mental illness or substance abuse, deemed by the investigator(s) to likely interfere with the patient's ability to sign informed consent, cooperate and participate in the study, or interfere with the interpretation of the results * Patients with a prior history of malignancies, other than multiple myeloma, are excluded unless the subject has been free of the disease for ≥ 5 years with the exception of the following non-invasive malignancies: * Basal cell carcinoma of the skin * Squamous cell carcinoma of the skin * Carcinoma in situ of the cervix * Carcinoma in situ of the breast * Incidental histological finding of prostate cancer (T1a or T1b using the TNM \[tumor, nodes, metastasis\] clinical staging system) or prostate cancer that is curative. * Patients with malabsorption or any other condition that in the opinion of the principal investigator could cause difficulty in absorption of drug * Patients that have previously progressed on pomalidomide treatment.