  If gravity is an emergent phenomenon, as suggested by several recent results,
then the structure of the action principle for gravity should encode this fact.
With this motivation we study several features of the Einstein-Hilbert action
and establish direct connections with horizon thermodynamics. We begin by
introducing the concept of holographically conjugate variables (HCVs) in terms
of which the surface term in the action has a specific relationship with the
bulk term. In addition to g_{ab} and its conjugate momentum \sqrt{-g} M^{cab},
this procedure allows us to (re)discover and motivate strongly the use of
f^{ab}=\sqrt{-g}g^{ab} and its conjugate momentum N^c_{ab}. The gravitational
action can then be interpreted as a momentum space action for these variables.
We also show that many expressions in classical gravity simplify considerably
in this approach. For example, the field equations can be written in a form
analogous to Hamilton's equations for a suitable Hamiltonian if we use these
variables. More importantly, the variation of the surface term, evaluated on
any null surface which acts a local Rindler horizon can be given a direct
thermodynamic interpretation. The term involving the variation of the dynamical
variable leads to T\delta S while the term involving the variation of the
conjugate momentum leads to S\delta T. We have found this correspondence only
for the choice of variables (g_{ab}, \sqrt{-g} M^{cab}) or (f^{ab}, N^c_{ab}).
We use this result to provide a direct thermodynamical interpretation of the
boundary condition in the action principle, when it is formulated in a
spacetime region bounded by the null surfaces. We analyse these features from
several different perspectives and provide a detailed description, which offers
insights about the nature of classical gravity and emergent paradigm.
