This trial studies the best dose and side effects of combining abexinostat and pembrolizumab in patients with certain types of cancer that have spread. Abexinostat may stop tumor growth by blocking certain enzymes, while pembrolizumab helps the immune system attack cancer cells. Pembrolizumab has been used effectively in treating various cancers, including melanoma and lung cancer. The goal is to find a more effective treatment for patients whose tumors have not responded to previous therapies.