Finite-cutoff holographic thermodynamics

We construct the first self-consistent framework for holographic thermodynamics in finite-cutoff holography, thereby capturing gravitational thermodynamics beyond the anti-de Sitter (AdS) boundary. We formulate two distinct first laws of thermodynamics: for the gravitational bulk, a Schwarzschild-AdS black hole with a finite Dirichlet cutoff, and for its dual T 2 deformed conformal field theory (CFT) on the boundary. The central innovation is promoting the deformation parameter to a thermodynamic variable on the boundary, which maps precisely to the cutoff radius in the bulk. We demonstrate exact duality between these two thermodynamic laws, establishing a complete holographic dictionary. This framework reveals that the T 2 deformation never lowers the confinement-deconfinement phase transition temperature below that of the original undeformed CFT, which acts as a global minimum. These results offer insights into gravitational thermodynamics with boundary constraints and quantum gravity in finite spacetime regions.