We study the exploratory Hamilton--Jacobi--Bellman (HJB) equation arising from the entropy-regularized exploratory control problem, which was formulated by Wang, Zariphopoulou and Zhou (J. Mach. Learn. Res., 21, 2020) in the context of reinforcement learning in continuous time and space. We establish the well-posedness and regularity of the viscosity solution to the equation, as well as the convergence of the exploratory control problem to the classical stochastic control problem when the level of exploration decays to zero. We then apply the general results to the exploratory temperature control problem, which was introduced by Gao, Xu and Zhou (arXiv:2005.04057, 2020) to design an endogenous temperature schedule for simulated annealing (SA) in the context of non-convex optimization. We derive an explicit rate of convergence for this problem as exploration diminishes to zero, and find that the steady state of the optimally controlled process exists, which is however neither a Dirac mass on the global optimum nor a Gibbs measure.