  Einstein-Hilbert action and its natural generalizations to higher dimensions
(like the Lanczos-Lovelock action) have certain peculiar features. All of them
can be separated into a bulk and a surface term, with a specific
("holographic") relationship between the two, so that either term can be used
to extract information about the other. Further, the surface term leads to
entropy of the horizons on-shell. It has been argued in the past that these
features are impossible to understand in the conventional approach but find a
natural explanation if we consider gravity as an emergent phenomenon. We
provide further support for this point of view in this paper. We describe an
alternative decomposition of the Einstein-Hilbert action and Lanczos-Lovelock
action into a new pair of surface and bulk terms, such that the surface term
becomes Wald entropy on a horizon and the bulk term is the energy density
(which is the ADM Hamiltonian density for Einstein gravity). We show that this
new pair also obeys a holographic relationship and give a thermodynamic
interpretation to this relation in this context. Since the bulk and surface
terms, in this decomposition, are related to energy and entropy, the
holographic condition can be thought of as analogous to inverting the
expression for entropy given as a function of energy S = S(E,V) to obtain the
energy E = E(S,V) in terms of the entropy in a normal thermodynamic system.
Thus the holographic nature of the action allows us to relate the descriptions
of the same system in terms of two different thermodynamic potentials. Some
further possible generalizations and implications are discussed.
