Black hole thermodynamics from Iyer-Wald formalism with variable Newton's constant

Abstract: Recent advances in black hole thermodynamics have intensified efforts to investigate its thermodynamic framework analogues to that of classical thermal systems. Departing from the extended phase space approach which involves a variable cosmological constant, the restricted phase space formalism focusing on the variations of Newton's gravitational constant in black hole thermodynamics is proposed. This approach enables the construction of a self-consistent thermodynamic structure that features both the first law and the Euler relation. In this paper, we demonstrate that for any diffeomorphism-invariant gravitational theory, the Iyer-Wald formalism is applicable to restricted phase space thermodynamics and construct a self-consistent extensive black hole thermodynamic system. Our work thereby establishes the Iyer-Wald approach as the geometric foundation for restricted phase space black hole thermodynamics.