We have performed AC specific heat measurements on the filled skutterudite PrOs4Sb12 at low temperature and under high pressure. The double superconducting transition was followed up to 4.3GPa in a diamond-anvil cell with argon as a pressure-transmitting medium. At low pressure, the slope ∂Tc1/∂P is at least 20% smaller than ∂Tc2/∂P and both of them are consistent with previous thermal expansion measurements by Oeschler et al. However, above 1GPa the behavior of the two transitions is similar, with Tc1-Tc2 stabilizing around 200mK. So the superconducting phase diagram under pressure (as well as under magnetic field) does not imply an intrinsic double superconducting transition. We discuss the origin of the double transition in the light of the pressure dependence of the Tc's and propose an explanation for the disparity between the slopes of Tc at low pressure.