Nowadays, low temperature superconducting hot electron bolometers (LTSC HEBs) are performing really well in the submillimeter wave range. Planar technologies are feasible that can be used for combining bolometer and antenna structures. High temperature superconducting (HTSC) films could also become the next to be used, although it seems that HTSC HEBs cannot reach their LTSC counterparts because of different phonon dynamics. In some previous models, it has been shown, however, that HTSC HEBs should be able to compete with traditional photoconductive detectors in terms of sensitivity. As the substrate temperature was assumed to be constant in these models, we wish to introduce in this paper a more detailed formulation where an extra cooling process (through the substrate) is taken into account. In a first part, simple heat propagation through the substrate is simulated to confirm the theory. The whole detector performance is analyzed in a second part, through a three-temperature model, for different device critical dimensions and thermal parameters. The influence of these characteristics on the lower cutoff frequency and on the maximal responsivity is then discussed. Different ways to improve these detectors are also given.