Inclusion Criteria:
* Pathologically proven diagnosis of high grade (aka grade III or IV) glioma (anaplastic astrocytoma, anaplastic oligodendroglioma, glioblastoma, gliosarcoma)
* Prior radiation therapy and standard temozolomide; additional therapies for previous progressions are eligible (prior bevacizumab and Optune are allowed)
* Three or more months from the end of chemoradiotherapy or have biopsy or imaging consistent with disease progression
* 19 years of age or older (the age of consent in Nebraska)
* Fully recovered from any toxicity of prior therapy that, in the opinion of the investigator, could impact tolerance to the study drug
* Eastern Cooperative Oncology Group (ECOG) Performance Status of 0-2
* Adequate bone marrow reserve (ANC count ≥1,500/mm3, hemoglobin \> 8 g/dL, platelet count ≥100,000/mm3)
* Adequate renal function (a serum creatinine that is at or below 2.0 mg/dL)
* Adequate hepatic function (serum AST and ALT less than 1.5 times the upper limits of normal, serum alkaline phosphatase less than 2.5 times the upper limits of normal)
* Able to provide written, informed consent
* Females of child-bearing potential must have a negative pregnancy test within 7 days of initiating study (non-child bearing potential is defined as age 55 years or older and no menses for two years or any age with surgical removal of the uterus and/or both ovaries)
* Females of reproductive potential must agree to employ an effective barrier method of birth control throughout the study and up to 6 months following treatment
Exclusion Criteria:
* Any life-threatening illness, medical condition, or organ system dysfunction which, in the investigator's opinion, could compromise the subject's safety, interfere with the absorption or metabolism of oral PCI-24781/Abexinostat, or put the study outcomes at undue risk
* Significant cardiovascular disease such as uncontrolled or symptomatic arrhythmia, congestive heart failure, or myocardial infarction within 6 months of screening, or any Class 3 or 4 cardiac disease as defined by the New York Heart Association Functional Classification
* Malabsorption syndrome, disease significantly affecting gastrointestinal function, or resection of the stomach or small bowel or ulcerative colitis, symptomatic inflammatory bowel disease, or partial or complete bowel obstruction
* Immunotherapy, chemotherapy, radiotherapy, corticosteroids (at dosages equivalent to prednisone \> 20 mg/day) or experimental therapy (other than PCI-24781/Abexinostat PO) within 4 weeks before first dose of study drug
* Concurrent use of enzyme-inducing antiepileptic drugs (phenytoin, phenobarbital, carbamazepine, felbamate, topiramate and oxcarbazepine)
* Any other active malignancy other than nonmelanoma skin cancer or controlled prostate cancer
* Known history of Human Immunodeficiency Virus (HIV) or active infection with Hepatitis C Virus (HCV) or Hepatitis B Virus (HBV) or any uncontrolled active systemic infection (no testing is required for eligibility)
* Creatinine \> 1.5 x institutional upper limit of normal (ULN); total bilirubin \> 1.5 x ULN (unless from Gilbert's disease), and aspartate aminotransferase (AST) or alanine aminotransferase (ALT) \> 2.5 x ULN
* Pregnant or breast-feeding
* Baseline ECG duration of the ventricular action potential corrected for heart rate (QTc interval) prolongation based on Fridericia's formula is \> 450 ms in males and \> 470 ms in females
* Concomitant valproic acid use, or another histone deacetylases (HDAC) inhibitor
* Receiving treatment with following medications and unable to discontinue treatment or switch medications prior to study enrollment:
* Amiodarone (Cordarone, Pacerone)
* Arsenic trioxide (Trisenox)
* Chlorpromazine (Aralen)
* Cisapride (Propulsid)
* Clarithromycin (Biaxin)
* Disopyramide (Norpace)
* Dofetilide (Tikosyn)
* Doperidol (Inapsine)
* Erythromycin (EryTab, Erythrocin)
* Flecanide (Tambocor)
* Haloperidol (Haldol)
* Ibutilide (Corvert)
* Methadone (Methadose, Dolophine)
* Moxifloxacin (Avelox)
* Pentamidine (Pentam, Nebupent)
* Pimozide (Orap)
* Procainamide (Procan, Pronestyl)
* Quinidine (Cardioquin, Quinaglute)
* Sotalol (Betapace)
* Thioridazine (Mellaril)
* Vandetanib (Zactima)
