We present ab initio (LDA + U sc ) studies of high-temperature and high-pressure elastic properties of pure as well as iron-bearing (ferrous, Fe 2+ , and ferric, Fe 3+ ) and aluminum-bearing MgSiO 3 postperovskite, the likely dominant phase in the deep lower mantle of the Earth. Thermal effects are addressed within the quasiharmonic approximation by combining vibrational density of states and static elastic coefficients. Aggregate elastic moduli and sound velocities for the Mg end members are successfully compared with scarce experimental data available. Effects of iron (Fe) and aluminum (Al) substitutions on elastic properties and their pressure and temperature dependence have been thoroughly investigated. At the observed perovskite to postperovskite transition (P = 125 GPa and T = 2,500 K), compressional and shear velocities increase by 0–1% and 1.5–3.75%, respectively. This observation is consistent with some seismic studies of the D ′′ discontinuity beneath the Caribbean, which suggests that our robust estimates of elastic properties of the postperovskite phase will be very helpful to understand lateral velocity variations in the deep lower mantle region and to constrain its composition and thermal structure.