Popular parametric and semiparametric hazards regression models for clustered survival data are inappropriate and inadequate when the unknown effects of different covariates and clustering are complex. For some survival studies involving time to occurrence of some asymptomatic events, survival times are typically interval censored. We develop a semiparametric hazards regression model by modeling the hazard function as a product of a parametric baseline hazard function and a nonparametric component that uses SBART (Soft Bayesian Additive Regression Trees or SBART (Linero and Yang, 2018)) to incorporate clustering, unknown functional forms of the main effects, and interaction effects of various covariates. In addition to being applicable for left, right, and interval-censored survival data, our methodology is implemented using a data augmentation scheme which allows for existing Bayesian backfitting algorithms to be used. We illustrate the practical implementation and advantages of our method via simulation studies and an analysis of a prostate cancer surgery study.