  We analyze the stellar ages obtained from a combination of Lick indices in
Borghi et al. for 140 massive and passive galaxies selected in the LEGA-C
survey at $0.6<z<0.9$. From their median age--redshift relation, we derive a
new direct measurement of $H(z)$ without any cosmological model assumption
using the cosmic chronometer approach. We thoroughly study the main systematics
involved in this analysis: the choice of the Lick indices combination, the
binning method, the assumed stellar population model, and the adopted star
formation history; these effects are included in the total error budget. We
obtain $H(z=0.75)= 98.8\pm33.6~\mathrm{km\,s^{-1}\,Mpc^{-1}}$. In parallel, we
also propose a simple framework based on a cosmological model to describe the
age--redshift relations in the context of galaxy downsizing. This allows us to
derive constraints on the Hubble constant $H_0$ and the typical galaxy
formation time. This new $H(z)$ measurement, whose accuracy is currently
limited by the scarcity of the sample analyzed, paves the road for the joint
study of the stellar populations of individual passive galaxies and the
expansion history of the Universe in light of future spectroscopic surveys.
