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We consider the universal sector of a $d$-dimensional large-$N$strongly-interacting holographic CFT on a black hole spacetime background $B$.When our CFT$_d$ is coupled to dynamical Einstein-Hilbert gravity with Newtonconstant $G_{d}$, the combined system can be shown to satisfy a version of thethermodynamic Generalized Second Law (GSL) at leading order in $G_{d}$. Thequantity $S_{CFT} + \frac{A(H_{B, \text{perturbed}})}{4G_{d}}$ isnon-decreasing, where $A(H_{B, \text{perturbed}})$ is the (time-dependent) areaof the new event horizon in the coupled theory. Our $S_{CFT}$ is the notion of(coarse-grained) CFT entropy outside the black hole given by causal holographicinformation -- a quantity in turn defined in the AdS$_{d+1}$ dual by therenormalized area $A_{ren}(H_{\rm bulk})$ of a corresponding bulk causalhorizon. A corollary is that the fine-grained GSL must hold for finiteprocesses taken as a whole, though local decreases of the fine-grainedgeneralized entropy are not obviously forbidden. Another corollary, given bysetting $G_{d} = 0$, states that no finite process taken as a whole canincrease the renormalized free energy $F = E_{out} - T S_{CFT} - \Omega J -\Phi Q$, with $T, \Omega, \Phi$ constants set by ${H}_B$. This latter corollaryconstitutes a 2nd law for appropriate non-compact AdS event horizons.

A coarse-grained generalized second law for holographic conformal field theories