  The connection between bulk and boundary thermodynamics in Einstein-Maxwell
theory is well established using AdS/CFT correspondence. In the context of
general higher derivative gravity coupled to a U(1) gauge field, we examine the
resemblance of the first law of thermodynamics between bulk and boundary,
followed by an extended phase space description on both sides. Higher
derivative terms related to different powers of the string theory parameter
$\alpha'$ emerged from a consistent truncation in the bulk supergravity action.
We demonstrate that one must include the fluctuation of $\alpha'$ in the bulk
thermodynamics as a bookkeeping tool to match the bulk first law and Smarr
relation with the boundary side. Consequently, the Euler relation and the
boundary first law are altered by adding two central charges ($\mathtt{a}$,
$\mathtt{c}$). To support our general conclusion, we consider the black hole in
Gauss-Bonnet gravity and the general four-derivative theory. Finally, we
examine the bulk and boundary aspects of the extended phase space description
for higher derivative corrected black holes.
