This article investigates the long-time behavior of conservative affine processes on the cone of symmetric positive semidefinite d × d matrices. In particular, for conservative and subcritical affine processes we show that a finite log-moment of the state-independent jump measure is sufficient for the existence of a unique limit distribution. Moreover, we study the convergence rate of the underlying transition kernel to the limit distribution: first, in a specific metric induced by the Laplace transform, and second, in the Wasserstein distance under a first moment assumption imposed on the state-independent jump measure and an additional condition on the diffusion parameter