Black hole thermodynamics in natural variables: the BTZ case

It has been recently shown for a wide range of black hole solutions in 4 and 5 dimensions that it is useful to reorganize the conventional thermodynamics on the inner and outer horizons in terms of left- and right-moving variables. The insight of arXiv:2310.13437 regarding the split of variables was loosely inspired by the example of the BTZ black hole and the corresponding dual CFT$_2$ thermodynamics governed by the Cardy formula. Here we revisit the BTZ case and its generalizations in 3d gravity theories with higher derivative corrections, and show formally the applicability of the newly defined variables and their direct relation to the Cardy formula, which is not a priori guaranteed.