Type Ia supernova (SN Ia) cosmology provides the most direct evidence for the presence of dark energy. This result is based on the assumption that the lookback time evolution of SN Ia luminosity, after light curve corrections, would be negligible. Recent studies show, however, that the Hubble residual (HR) of SN Ia is correlated with the mass and morphology of host galaxies, implying the possible dependence of SN Ia luminosity on host galaxy properties. In order to investigate this more directly, we have initiated a spectroscopic survey for early-type host galaxies, for which population age and metallicity can be more reliably determined from the absorption lines. In this first paper of the series, we present here the results from high signal-to-noise ratio (≳100 per pixel) spectra for 27 nearby host galaxies in the southern hemisphere. For the first time in host galaxy studies, we find a significant (∼3.9σ) correlation between host galaxy mass (velocity dispersion) and population age, which is consistent with the "downsizing" trend among non-host early-type galaxies. This result is rather insensitive to the choice of population synthesis models. Since we find no correlation with metallicity, our result suggests that stellar population age is mainly responsible for the relation between host mass and HR. If confirmed, this would imply that the luminosity evolution plays a major role in the systematic uncertainties of SN Ia cosmology.