  We study thermodynamics of black hole solutions in Lanczos-Lovelock AdS
gravity in d+1 dimensions coupled to nonlinear electrodynamics and a
Stueckelberg scalar field. This class of theories is used in the context of
gauge/gravity duality to describe a high-temperature superconductor in d
dimensions. Larger number of coupling constants in the gravitational side is
necessary to widen a domain of validity of physical quantities in a dual QFT.
We regularize the gravitational action and find the finite conserved quantities
for a planar black hole with scalar hair. Then we derive the quantum
statistical relation in the Euclidean sector of the theory, and obtain the
exact formula for the free energy of the superconductor in the holographic
quantum field theory. Our result is analytic and it includes the effects of
backreaction of the gravitational field. We further discuss on how this formula
could be used to analyze second order phase transitions through the
discontinuities of the free energy, in order to classify holographic
superconductors in terms of the parameters in the theory.
