  The deep connection between gravitational dynamics and horizon thermodynamics
leads to several intriguing features both in general relativity and in
Lanczos-Lovelock theories of gravity. Recently in arXiv:1312.3253 several
additional results strengthening the above connection have been established
within the framework of general relativity. In this work we provide a
generalization of the above setup to Lanczos-Lovelock gravity as well. To our
expectation it turns out that most of the results obtained in the context of
general relativity generalize to Lanczos-Lovelock gravity in a straightforward
but non-trivial manner.
  Another very interesting feature for gravity is that gravitational field
equations for arbitrary static and spherically symmetric spacetimes with
horizon can be written as a thermodynamic identity in the near horizon limit.
This result holds in both general relativity and in Lanczos-Lovelock gravity as
well. In a previous work [arXiv:1505.05297] we have shown that, for an
arbitrary spacetime, the gravitational field equations near any null surface
generically leads to a thermodynamic identity. In this work, we have also
generalized this result to Lanczos-Lovelock gravity by showing that
gravitational field equations for Lanczos-Lovelock gravity near an arbitrary
null surface can be written as a thermodynamic identity. By taking appropriate
limit to general relativity we can reproduce the results presented in
arXiv:1312.3253 and arXiv:1505.05297.
