No Hawking-Page phase transition in three dimensions

We investigate whether or not the Hawking-Page phase transition is possible to occur in three dimensions. Starting with the simplest class of Lanczos-Lovelock action, thermodynamic behavior of all AdS-type black holes without charge falls into two classes: Schwarzschild-AdS black holes in even dimensions and Chern-Simons black holes in odd dimensions. The former class can provide the Hawking-Page transition between Schwarzschild-AdS black holes and thermal AdS space. On the other hand, the latter class is exceptional and thus the Hawking-Page transition is hard to occur. In three dimensions, a second-order phase transition might occur between the non-rotating BTZ black hole and the massless BTZ black hole (thermal AdS space), instead of the first-order Hawking-Page transition between the non-rotating BTZ black hole and thermal AdS space.