  Recently, it has been shown that for out-of-equilibrium systems, there are
additional constraints on thermodynamical evolution besides the ordinary second
law. These form a new family of second laws of thermodynamics, which are
equivalent to the monotonicity of quantum R\'enyi divergences. In black hole
thermodynamics, the usual second law is manifest as the area increase theorem.
Hence one may ask if these additional laws imply new restrictions for
gravitational dynamics, such as for out-of-equilibrium black holes? Inspired by
this question, we study these constraints within the AdS/CFT correspondence.
First, we show that the R\'enyi divergence can be computed via a Euclidean path
integral for a certain class of excited CFT states. Applying this construction
to the boundary CFT, the R\'enyi divergence is evaluated as the renormalized
action for a particular bulk solution of a minimally coupled gravity-scalar
system. Further, within this framework, we show that there exist transitions
which are allowed by the traditional second law, but forbidden by the
additional thermodynamical constraints. We speculate on the implications of our
findings.
