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The anomalous conformal transformation law of the generalized entropy isfound for dilaton gravity coupled to a 1+1 conformal matter sector with centralcharges $c = \tilde{c}$. (When $c \ne \tilde{c}$ the generalized entropy is notinvariant under local Lorentz boosts.) It is shown that a certain second nullderivative of the entropy, $S_\text{gen}" + (6/c)(S_\text{out}')^2$, isprimary, and therefore retains its sign under a general conformaltransformation. Consequently all conformal vacua have increasing entropy oncausal horizons. Alternative definitions of the horizon, including apparent ordynamical horizons, can have decreasing entropy in any dimension $D \ge 2$.This indicates that the generalized second law should be defined using thecausal horizon.

Testing the Generalized Second Law in 1+1 dimensional Conformal Vacua: An Argument for the Causal Horizon