  Here we consider the generalized Oppenheimer-Snyder collapse of a star into a
four-dimensional Einstein-Gauss-Bonnet black hole as well as a class of regular
black holes labeled by the polytropic index of the stellar matter. We then
analyze the nature of the horizon and the corresponding surface gravity outside
and inside the star. The Hayward and Nielsen-Visser dynamical surface gravity
are in agreement with the one resulting from the Killing vector of the outer
static metric. However, these two definitions inside the star do not coincide
with the Killing surface gravity outside the star when the star crosses the
event horizon. This motivates us to study the surface gravity using Fodor's
approach to have a unique surface gravity at the mentioned moment. Then the
extremality condition and the first law of thermodynamics are discussed at the
trapping horizon of the star.
