We experimentally investigate the relaxation of spatially confined microcavity polaritons. We measure the time- and energy-resolved photoluminescence under resonant excitation and in the low-density regime. In this way, we have access to the time evolution of the energy distribution of the polariton population. We show that, when one confined level is resonantly excited, after an initial transient, the population of the confined levels is thermally distributed. The reported efficiency of the relaxation process strongly depends on the confinement size. These experimental findings are well reproduced by a theoretical model accounting for the coupling between the confined states and a bath of acoustic phonons. Our results thus suggest that the phonon-mediated relaxation mechanisms are enhanced in the presence of spatial confinement.