Heat and work in black hole thermodynamics via holography

We propose a formulation of black hole thermodynamics that incorporates the notions of heat and work, based on the thermodynamics in quantum theory and the AdS/CFT correspondence. First, for coupled holographic CFTs, we define a coarse-graining procedure adopting the principle of maximum entropy. Employing this approach, when the system is divided into a target system and thermal baths, we formulate the first and second laws, as well as the fundamental thermodynamic relation. Then, by translating the resulting thermodynamics into the AdS gravity language, we construct a thermodynamic framework for composite black hole systems that encompasses both heat and work. This formulation relies on holography, but not on energy conditions on the gravity side. We also argue that the second law serves as a necessary criterion for the UV completeness of gravitational theories.