Finite-cutoff Holographic Thermodynamics

We develop a framework for holographic thermodynamics in finite-cutoff holography, extending the anti-de Sitter/conformal field theory (AdS/CFT) correspondence to incorporate a finite radial cutoff in the bulk and a $T^2$-deformed CFT on the boundary. We formulate the first laws of thermodynamics for a Schwarzschild-AdS (SAdS) black hole with a Dirichlet cutoff on the quasilocal boundary and its dual deformed CFT, introducing the deformation parameter as a thermodynamic variable. The holographic Euler relation for the deformed CFT and its equation of state are derived, alongside the Smarr relation for the bulk. We show that the Rupert teardrop coexistence curve defines a phase space island where deformation flow alters states, with up to three deformed CFTs or cut-off SAdS sharing a same phase transition temperature, one matching the seed CFT or original SAdS. These results offer insights into gravitational thermodynamics with boundary constraints and quantum gravity in finite spacetime regions.