Multiferroic materials, in which ferroelectric and magnetic ordering coexist, are of fundamental interest for the development of multi-state memory devices that allow for electrical writing and non-destructive magnetic readout operation. The great challenge is to create multiferroic materials that operate at room temperature and have a large ferroelectric polarization P. Cupric oxide, CuO, is promising because it exhibits a significant polarization, that is, P~0.1 μC cm(-2), for a spin-spiral multiferroic. Unfortunately, CuO is only ferroelectric in a temperature range of 20 K, from 210 to 230 K. Here, by using a combination of density functional theory and Monte Carlo calculations, we establish that pressure-driven phase competition induces a giant stabilization of the multiferroic phase of CuO, which at 20-40 GPa becomes stable in a domain larger than 300 K, from 0 to T>300 K. Thus, under high pressure, CuO is predicted to be a room-temperature multiferroic with large polarization.