In the ideal pressure conditions of a helium-filled diamond anvil cell, we have been able to probe the resistivity and, for the first time, the specific heat of the heavy fermion superconductor CeCu2Si2 at pressures over 6 GPa, down to temperatures below 100 mK, and in a magnetic field up to 8 T. We clearly observed the superconducting jump using the AC calorimetry technique, which provides a semi-quantitative measure of the sample specific heat. The evolution of the superconducting transition with pressure was observed quasi-simultaneously in a single sample in both the resistivity and heat capacity. The jump in Cp hints at changes in the coupling regime. When Tc is a rapidly varying function of pressure, the resistive transition is broadened and strongly dependent on the measuring current. When Tc has a maximum at 2K, the residual resistivity shows a peak, and the resistivity is linear in temperature above Tc