Spectra of the hydrated electron in pressurized light and heavy water at temperatures up to and beyond the critical temperature are reported, for wavelengths between 0.4 and 1.7 mum. In agreement with previous work, spectra can be approximately represented by a Gaussian function on the low-energy side, and a Lorentzian function on the high-energy side in subcritical water, but deviations from this form are very clear above 200 degreesC. The spectrum shifts strongly to the red as temperature rises. At supercritical temperatures, the spectrum shifts slightly to the red as density decreases, and the Gaussian - Lorentzian form is a very poor description. Application of spectral moment theory allows one to make an estimate of the average size of the electron wave function and of its kinetic energy. It appears that for water densities below about 0.6 g/cc, and down to below 0.1 g/cc, the average radius of gyration for the electron remains constant at around 3.4 Angstrom, and its absorption maximum is near 0.9 eV. For higher densities, the electron is squeezed into a smaller cavity and the spectrum is shifted to the blue. The enthalpy and free energy of electron hydration are derived as a function of temperature on the basis of existing equilibrium data and absolute proton hydration energies derived from the cluster-based common point method. In a discussion, we compare the effective "size" of the hydrated electron derived from both methods.