CuO, known to be multiferroic (MF) from T-L = 213 K to T-N = 230 K at ambient pressure, has been the subject of debates about its ability to exhibit multiferroicity at room temperature (RT) under high hydrostatic pressure. Here we address this question based on theoretical and experimental investigations. The influence of hydrostatic pressure on T-L and T-N has been estimated from ab initio calculations combined with classical Monte-Carlo simulations and a quasi-1D antiferromagnetic analytical model. From the experimental side, electric permittivity anomalies related to ferroelectric transitions have been followed with dielectric measurements on single crystals up to 6.1 GPa. We show that the temperature T-N below which the MF state forms increases with pressure linearly to higher pressure that hitherto supposed, and indeed based on our calculations, should exceed RT above about 20 GPa.