  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.
