Mocetinostat is an orally administered histone deacetylase (HDAC) inhibitor. This study is a Phase 2 trial evaluating the efficacy of mocetinostat in patients that have advanced urothelial carcinoma that has specific changes in tumor genes. Patients must have previously received treatment with chemotherapy that included a "platinum-containing agent" such as cisplatin. The study will enroll in stages, with 15 patients in the first stage. More patients will be added to the study if enough patients having beneficial responses are observed. Mocetinostat will be administered using oral capsules three times each week (eg, Monday, Wednesday and Friday). The study is designed to evaluate whether the number of patients responding to treatment is substantially higher than would be expected with other available treatments.