BTZ black hole with higher derivatives, the second law of thermodynamics, and statistical entropy: A new proposal

I consider the thermodynamics of the BTZ black hole in the presence of the higher curvature and gravitational Chern-Simons terms, and its statistical entropy. I propose a new thermodynamical entropy, which is manifestly non-negative, such that the second law of thermodynamics is satisfied. I show that the new thermodynamical entropy agrees perfectly with the statistical entropy for all the values of the conformal factor of the higher curvature terms and the coupling constant of the gravitational Chern-Simons term, in contrast to some disagreements in the literature. The agreement with both the higher curvature and gravitational Chern-Simons terms is possible because of an appropriate balancing of them, though it is not a trivial matter because of a conflict in the appropriate Hilbert space for a well-defined conformal field theory for each term.