  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.
