We present the discovery of the transiting exoplanets HAT-P-65b and HAT-P-66b, with orbital periods of 2.6055 d and 2.9721 d, masses of $0.527 pm 0.083$ M$_{J}$ and $0.783 pm 0.057$ M$_{J}$ and inflated radii of $1.89 pm 0.13$ R$_{J}$ and $1.59^{+0.16}_{-0.10}$ R$_{J}$, respectively. They orbit moderately bright ($V=13.145 pm 0.029$, and $V=12.993 pm 0.052$) stars of mass $1.212 pm 0.050$ M$_{odot}$ and $1.255^{+0.107}_{-0.054}$ M$_{odot}$. The stars are at the main sequence turnoff. While it is well known that the radii of close-in giant planets are correlated with their equilibrium temperatures, whether or not the radii of planets increase in time as their hosts evolve and become more luminous is an open question. Looking at the broader sample of well-characterized close-in transiting giant planets, we find that there is a statistically significant correlation between planetary radii and the fractional ages of their host stars, with a false alarm probability of only 0.0041%. We find that the correlation between the radii of planets and the fractional ages of their hosts is fully explained by the known correlation between planetary radii and their present day equilibrium temperatures, however if the zero-age main sequence equilibrium temperature is used in place of the present day equilibrium temperature then a correlation with age must also be included to explain the planetary radii. This suggests that, after contracting during the pre-main-sequence, close-in giant planets are re-inflated over time due to the increasing level of irradiation received from their host stars. Prior theoretical work indicates that such a dynamic response to irradiation requires a significant fraction of the incident energy to be deposited deep within the planetary interiors.
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