The core collapse of a massive star results in the formation of a proto-neutron star (PNS). If enough material is accreted onto a PNS it will become gravitationally unstable and further collapse into a black-hole (BH). We perform a systematic study of failing core-collapse supernovae in spherical symmetry for a wide range of presupernova progenitor stars and equations of state (EOSs) of nuclear matter. We analyze how variations in progenitor structure and the EOS of dense matter above nuclear saturation density affect the PNS evolution and subsequent BH formation. Comparisons of core-collapse for a given progenitor star and different EOSs show that the path traced by the PNS in mass-entropy phase space $M_{mathrm{grav}}^{mathrm{PNS}}-tilde{s}$ is well correlated with the progenitor compactness and almost EOS independent, apart from the final endpoint. Furthermore, BH formation occurs, to a very good approximation, soon after the PNS overcomes the maximum textit{gravitational} mass supported by a hot NS with constant entropy equal to $tilde{s}$. These results show a path to constraining the temperature dependence of the EOS through the detection of neutrinos from a failed galactic supernova.
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