The CaSnO3 perovskite is investigated under geochemical pressure, up to 25 GPa, by means of periodic ab initio calculations performed at B3LYP level with local Gaussian-type orbital basis sets. Structural, elastic, and spectroscopic (phonon wave-numbers, infrared and Raman intensities) properties are fully characterized and discussed. The evolution of the Raman spectrum of CaSnO3 under pressure is reported to remarkably agree with a recent experimental determination [J. Kung, Y. J. Lin, and C. M. Lin, J. Chem. Phys. 135, 224507 (2011)] as regards both wave-number shifts and intensity changes. All phonon modes are symmetry-labeled and bands assigned. The single-crystal total spectrum is symmetry-decomposed into the six directional spectra related to the components of the polarizability tensor. The infrared spectrum at increasing pressure is reported for the first time and its main features discussed. All calculations are performed using the Crystal14 program, taking advantage of the new implementation of analytical infrared and Raman intensities for crystalline materials.