We predict the phase diagram of CaSiO3 perovskite, finding the tetragonal I4/mcm structure transforming to cubic Pm (3) over barm with increasing temperature. The transition temperature is 1150 K at 0 GPa, and 2450 K at 140 GPa. The c/a ratio of the tetragonal structure is 1.018 at 100 GPa and increases on compression, as does the static enthalpy difference between tetragonal and cubic structures. The elastic constants of the tetragonal phase at static conditions differ substantially from those of the cubic phase with the Voigt-Reuss-Hill shear modulus 29% less at 100 GPa. Computations are based on density functional theory in the local density and generalized gradient approximations. The phase diagram and high temperature elastic constants are computed using a mean field theory with parameters of the Landau potential determined via structurally constrained density functional theory calculations. We present a simple scheme for systematically searching for the ground state over all perovskite structures derivable from octahedral rotations within the context of symmetry-preserving relaxation, which confirms tetragonal I4/mcm as the ground state in density functional theory. We argue that the experimental x-ray diffraction pattern can be explained by the I4/mcm phase by considering the development of preferred orientation under uniaxial compression.