A key difference between black holes and ordinary thermal systems is the absence of a pressure-volume work term in the first law of black hole thermodynamics. It is possible to introduce such a work term for black holes in backgrounds with a cosmological constant by treating the cosmological constant as a pressure, offering a rich gravitational perspective on everyday phenomena. Missing, however, is justification for allowing variations of the cosmological constant. In this talk I will present a higher-dimensional origin of 'extended black hole thermodynamics' using holographic braneworlds. In this set-up, gravity is coupled to a lower-dimensional brane such that classical black holes in a bulk anti-de Sitter spacetime correspond to exact quantum corrected black holes localized on the brane, including all orders of semi-classical backreaction. Crucially, varying the tension of the brane leads to a dynamical cosmological constant on the brane, and, correspondingly, a variable pressure attributed to the brane black hole. In other words, standard thermodynamics of classical black holes induces extended thermodynamics of `quantum' black holes on a brane. As proof of concept, I will present the extended thermodynamics of the quantum BTZ black hole, also providing a microscopic interpretation using `double holography’.