This work is concerned with the stability, existence, and uniqueness of invariant measure for a hybrid diffusion. Under new conditions, it is shown that the hybrid diffusion possesses a unique invariant measure and its transition probability converges exponentially fast to its invariant measure under a Wasserstein distance. For the discretized process, it is demonstrated that similar results are obtained when the time step size is sufficiently small. As a result, it is shown that the invariant measure of the path-dependent hybrid diffusion can be approximated by that of the discretized process.