We report optical and near-infrared broad band observations of the short-duration GRB 050724 host galaxy, used to construct its spectral energy distribution (SED). Unlike the hosts of long-duration gamma-ray bursts (GRBs), which show younger stellar populations, the SED of the GRB 050724 host galaxy is optimally fitted with a synthetic elliptical galaxy template based on an evolved stellar population (age ~2.6 Gyr). The SED of the host is difficult to reproduce with non-evolving metallicity templates. In contrast, if the short GRB host galaxy metallicity enrichment is considered, the synthetic templates fit the observed SED satisfactorily. The internal host extinction is low ($A_{\rm v} \lesssim 0.4$ mag) so it cannot explain the faintness of the afterglow. This short GRB host galaxy is more massive (${∼}5\times10^{10}\, M_{⊙}$) and luminous (${∼}1.1~ L^{⋆}$) than most of the long-duration GRB hosts. A statistical comparison based on the ages of short- and long-duration GRB host galaxies strongly suggests that short-duration GRB hosts contain, on average, older progenitors. These findings support a different origin for short- and long-duration GRBs.