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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:msup> <a:mi>T</a:mi> <a:mn>2</a:mn> </a:msup> </a:math> 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 <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:msup> <b:mi>T</b:mi> <b:mn>2</b:mn> </b:msup> </b:math> 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.