The present invention relates to a crosslinked rubber composition based on at least one elastomer of ethylene-propylene-diene terpolymer type (EPDM) and on at least one phase-change material (PCM), to a process for preparing this composition and to a multilayer pipe incorporating it. This composition includes at least 100 phr (phr: parts by weight per one hundred parts of elastomer(s)) of at least one phase-change material (PCM), and it is such that said at least one PCM is dispersed in the crosslinked composition and is provided with protection means that are capable of preventing its dispersion therein at a temperature above its melting point, which composition has a breaking strength of greater than 3 MPa and/or an elongation at break of greater than 100%, these properties being measured at 23° C. according to standard ASTM D 412.