  This work introduces a novel prescription for the expression of the
thermodynamic potentials associated with the couplings of a Lanczos-Lovelock
theory. These potentials emerge in theories with multiple couplings, where the
ratio between them provide intrinsic length scales that break scale invariance.
Our prescription, derived from the covariant phase space formalism, differs
from previous approaches by enabling the construction of finite potentials
without reference to any background. To do so, we consistently work with
finite-size systems with Dirichlet boundary conditions and rigorously take into
account boundary and corner terms: including these terms is found to be crucial
for relaxing the integrability conditions for phase space quantities that were
required in previous works. We apply this prescription to the first law of
(extended) thermodynamics for stationary black holes, and derive a version of
the Smarr formula that holds for static black holes with arbitrary asymptotic
behaviour.
