Weak electron-phonon interaction in metals at low temperatures forms the basis of operation for cryogenic hot-electron bolometers and calorimeters. Here, we develop a thermometry scheme based on proximity supercurrent to study the thermal response of a thin gold film on a SiO 2 platform at temperatures below 100 mK. We find that the exponent of the power law describing electron-phonon coupling in the film drops by approximately 1 / 2 as the platform is micromachined and released from its substrate. This contrasts the conventional theory for bulklike geometries that predicts integer-valued exponents. We attribute the fractional change to a modified phonon spectrum described by recent theoretical developments.