Thermodynamic route to field equations in Lanczos-Lovelock gravity

Spacetimes with horizons show a resemblance to thermodynamic systems and one can associate the notions of temperature and entropy with them. In the case of Einstein-Hilbert gravity, it is possible to interpret Einstein's equations as the thermodynamic identity $TdS=dE+PdV$ for a spherically symmetric spacetime and thus provide a thermodynamic route to understand the dynamics of gravity. We study this approach further and show that the field equations for the Lanczos-Lovelock action in a spherically symmetric spacetime can also be expressed as $TdS=dE+PdV$ with S and E given by expressions previously derived in the literature by other approaches. The Lanczos-Lovelock Lagrangians are of the form L=QₐᵇᶜᵈRᵃbcd with ∇bQₐᵇᶜᵈ=0. In such models, the expansion of Qₐᵇᶜᵈ in terms of the derivatives of the metric tensor determines the structure of the theory and higher order terms can be interpreted as quantum corrections to Einstein gravity. Our result indicates a deep connection between the thermodynamics of horizons and the allowed quantum corrections to standard Einstein gravity, and shows that the relation $TdS=dE+PdV$ has a greater domain of validity than Einstein's field equations.