  The variational problem of gravity theories is directly related to black hole
thermodynamics. For asymptotically locally AdS backgrounds it is known that
holographic renormalization results in a variational principle in terms of
equivalence classes of boundary data under the local asymptotic symmetries of
the theory, which automatically leads to finite conserved charges satisfying
the first law of thermodynamics. We show that this connection holds well beyond
asymptotically AdS black holes. In particular, we formulate the variational
problem for $\mathcal{N}=2$ STU supergravity in four dimensions with boundary
conditions corresponding to those obeyed by the so called `subtracted
geometries'. We show that such boundary conditions can be imposed covariantly
in terms of a set of asymptotic second class constraints, and we derive the
appropriate boundary terms that render the variational problem well posed in
two different duality frames of the STU model. This allows us to define finite
conserved charges associated with any asymptotic Killing vector and to
demonstrate that these charges satisfy the Smarr formula and the first law of
thermodynamics. Moreover, by uplifting the theory to five dimensions and then
reducing on a 2-sphere, we provide a precise map between the thermodynamic
observables of the subtracted geometries and those of the BTZ black hole.
Surface terms play a crucial role in this identification.

High Energy Physics - Theory;General Relativity and Quantum Cosmology