  Recently, the BTZ black hole in the presence of the gravitational
Chern-Simons (GCS) term has been studied and it has been found that the usual
thermodynamical quantities, like as the black hole mass, angular momentum, and
black hole entropy, are modified. But, for large values of the GCS coupling,
where the modification terms dominate the original terms, some exotic behaviors
occur, like as the roles of the mass and angular momentum are interchanged and
the black hole entropy depends more on the $inner$-horizon area than the outer
one. A basic physical problem of this system is that the form of entropy does
not guarantee the second law of thermodynamics, in contrast to the
Bekenstein-Hawking (BH) entropy. Moreover, this entropy does $not$ agree with
the statistical entropy, in contrast to a good agreement for small values of
the GCS coupling. Here I find that there is another entropy formula where the
usual BH form dominates the inner-horizon term again, as in the small GCS
coupling, such as the second law of thermodynamics can be guaranteed. I compare
the result of the holographic approach with the classical-
symmetry-algebra-based approach and I find exact agreements even with the
higher-derivative term of GCS. This provides a non-trivial check of the
AdS/CFT-correspondence in the presence of higher-derivative terms in the
gravity action.

High Energy Physics - Theory