  We consider the first law of black hole thermodynamics in an asymptotically
anti-de Sitter spacetime in the class of gravitational theories whose
gravitational Lagrangian is an arbitrary function of the Ricci scalar. We first
show that the conserved quantities in this class of gravitational theories
constructed through conformal completion remain unchanged under the conformal
transformation into the Einstein frame. We then prove that the mass and the
angular momenta defined by these conserved quantities, along with the entropy
defined by the Noether charge, satisfy the first law of black hole
thermodynamics, not only in Einstein gravity but also in the higher curvature
gravity within the class under consideration. We also point out that it is
naturally understood in the symplectic formalism that the mass satisfying the
first law should be necessarily defined associated with the timelike Killing
vector nonrotating at infinity. Finally, a possible generalization into a wider
class of gravitational theories is discussed.
