  In arXiv:1508.01343 [hep-th], one of the authors proposed that in AdS/CFT the
gravity dual of the boundary $c$-theorem is the second law of thermodynamics
satisfied by causal horizons in AdS and this was verified for Einstein gravity
in the bulk. In this paper we verify this for higher derivative theories. We
pick up theories for which an entropy expression satisfying the second law
exists and show that the entropy density evaluated on the causal horizon in a
RG flow geometry is a holographic c-function. We also prove that given a theory
of gravity described by a local covariant action in the bulk a sufficient
condition to ensure holographic c-theorem is that the second law of causal
horizon thermodynamics be satisfied by the theory. This allows us to explicitly
construct holographic c-function in a theory where there is curvature coupling
between gravity and matter and standard null energy condition cannot be defined
although second law is known to hold. Based on the duality between c-theorem
and the second law of causal horizon thermodynamics proposed in
arXiv:1508.01343 [hep-th] and the supporting calculations of this paper we
conjecture that every Unitary higher derivative theory of gravity in AdS
satisfies the second law of causal horizon thermodynamics. If this is not true
then c-theorem will be violated in a unitary Lorentz invariant field theory.
