Using elastic constants determined by ab initio molecular dynamics over a range of temperatures, we show that the seismic anisotropy of MgSiO₃ perovskite is significantly temperature dependent. At 90 GPa, the direction of greatest shear wave splitting changes from [001] at 0 K to [010] at 3500 K. In addition, there is no shear wave splitting in the [100] direction at 0 K, but a significant amount of splitting is exhibited at 3500 K. We have also calculated the first set of high-temperature elastic constants of the new post-perovskite phase by ab initio molecular dynamics methods. We find that, in contrast to MgSiO₃ perovskite, temperature has little effect on the acoustic anisotropy of this phase. We determine ∂ln V_S/∂ln V_P for the post-perovskite phase and find it to be very low, about 0.9. This may be an important parameter for mapping out the post-perovskite phase in the lowermost mantle.