We have performed density-functional theory calculations for three crystallographic phases (cuprite, CdI(2), and CdCl(2)) of the cuprous oxide by using both the local-density approximation (LDA) and the Perdew-Burke-Ernzerhof generalized-gradient approximation. The latter gives a very good description of the properties of the cuprite phase at room temperature. In particular, the bulk modulus and the elastic constants at zero pressure are in excellent agreement with experiment. At 10 GPa (7 in LDA calculations), the transition from the cuprite to the CdI(2) phase occurs, and the latter remains the phase having the smallest Gibbs energy up to the maximum pressure we have considered (20 GPa). We have also determined the elastic constants of Cu(2)O in the cuprite phase for various applied pressures. The results indicate that this structure becomes unstable with respect to trigonal deformations before the transition to the CdI(2) phase. On the other hand, no indication of instability with respect to tetragonal deformations has been found. This kind of instability would occur at pressures greater than the phase transition pressure.