The pressure–temperature phase diagrams of the heavy fermion antiferromagnet CeRhIn₅ and the heavy fermion superconductor CeCoIn₅ have been studied under hydrostatic pressure by ac calorimetry and ac susceptibility measurements using diamond anvil cells with argon as the pressure medium. In CeRhIn₅, the use of a highly hydrostatic pressure transmitting medium allows for a clean simultaneous determination by a bulk probe of the antiferromagnetic and superconducting transitions. We compare our new phase diagram with the previous ones, discuss the nature (first or second order) of the various lines, and the coexistence of antiferromagnetic order and superconductivity. The link between the collapse of the superconducting heat anomaly and the broadening of the antiferromagnetic transition points to an inhomogeneous appearance of superconductivity below P_c ≈ 1.95 GPa. Homogeneous bulk superconductivity is only observed above this critical pressure. We present a detailed analysis of the influence of pressure inhomogeneities on the specific heat anomalies which emphasizes that the observed broadening of the transitions near P_c is connected with the first-order transition. For CeCoIn₅ we show that the large specific heat anomaly observed at T_c at ambient pressure is suppressed linearly at least up to 3 GPa.