  We extend Padmanabhan's entropy functional formalism to show that, in
addition to the Gauss-Bonnet or the entire series of Lanczos-Lovelock
Lagrangians already obtained, more general higher-order corrections to General
Relativity, i.e., the so-called modified gravity theories, also emerge
naturally from this formalism. This extension shows that the formalism
constitutes a valuable tool to investigate, at each order in the curvature, the
possible structure the higher-order modified gravity theories might have. As an
application, the extended formalism is used to evaluate the horizon entropy in
a modified gravity theory of the second-order in the curvature. Our findings
are in agreement with previous results from the literature.
