Bayesian Additive Regression Trees (BART) is a popular Bayesian non-parametric regression model that is commonly used in causal inference and beyond. Its strong predictive performance is supported by well-developed estimation theory, comprising guarantees that its posterior distribution concentrates around the true regression function at optimal rates under various data generative settings and for appropriate prior choices. However, the computational properties of the widely-used BART sampler proposed by Chipman et al. (2010) are yet to be well-understood. In this paper, we perform an asymptotic analysis of a slightly modified version of the default BART sampler when fitted to data-generating processes with discrete covariates. We show that the sampler's time to convergence, evaluated in terms of the hitting time of a high posterior density set, increases with the number of training samples, due to the multi-modal nature of the target posterior. On the other hand, we show that this trend can be dampened by simple changes, such as increasing the number of trees in the ensemble or raising the temperature of the sampler. These results provide a nuanced picture on the computational efficiency of the BART sampler in the presence of large amounts of training data while suggesting strategies to improve the sampler. We complement our theoretical analysis with a simulation study focusing on the default BART sampler. We observe that the increasing trend of convergence time against number training samples holds for the default BART sampler and is robust to changes in sampler initialization, number of burn-in iterations, feature selection prior, and discretization strategy. On the other hand, increasing the number of trees or raising the temperature sharply dampens this trend, as indicated by our theory.