Entropy function from the gravitational surface action for an extremal near horizon black hole

It is often argued that {\it all the information of a gravitational theory is encoded in the surface term of the action}; which means one can find several physical quantities just from the surface term without incorporating the bulk part of the action. This has been observed in various instances; e.g. derivation of the Einstein's equations, surface term calculated on the horizon leads to entropy, etc. Here I investigate the role of it in the context of entropy function and entropy of extremal near horizon black holes. Considering only the Gibbons-Hawking-York (GHY) surface term to define an entropy function for the extremal near horizon black hole solution, it is observed that the extremization of such function leads to the exact value of the horizon entropy. This analysis again supports the previous claim that there exists a ``{\it holographic}'' nature in the gravitational action -- surface term contains the information of the bulk.

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