  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
Lanczos-Lovelock action in a spherically symmetric spacetime can also be
expressed as TdS = dE + PdV with S and E being given by expressions previously
derived in the literature by other approaches. The Lanczos-Lovelock Lagrangians
are of the form L=Q_a^{bcd}R^a_{bcd} with \nabla_b Q^{abcd}=0. In such models,
the expansion of Q^{abcd} in terms of the derivatives of the metric tensor
determines the structure of the theory and higher order terms can be
interpreted 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 that Einstein's field equations.
