A Note on Thermodynamics of Black Holes in Lovelock Gravity

The Lovelock gravity consists of the dimensionally extended Euler densities. The geometry and horizon structure of black hole solutions could be quite complicated in this gravity, however, we find that some thermodynamic quantities of the black holes like the mass, Hawking temperature and entropy, have simple forms expressed in terms of horizon radius. The case with black hole horizon being a Ricci flat hypersurface is particularly simple. In that case the black holes are always thermodynamically stable with a positive heat capacity and their entropy still obeys the area formula, which is no longer valid for black holes with positive or negative constant curvature horizon hypersurface. In addition, for black holes in the gravity theory of Ricci scalar plus a 2n-dimensional Euler density with a positive coefficient, thermodynamically stable small black holes always exist in D = 2n + 1 dimensions, which are absent in the case without the Euler density term, while the thermodynamic properties of the black hole solutions with the Euler density term are qualitatively similar to those of black holes without the Euler density term as D > 2n+ 1. ∗e-mail address: [email protected] Over the past years gravity with higher derivative curvature terms has received a lot of attention, in particular, in the brane world scenario and black hole thermodynamics. In the former, the motivation is to study the corrections of these higher derivative terms to the Newton law of gravity on the brane, or to avoid the singularity in bulk by using these higher derivative terms. In the field of black hole thermodynamics, it is expected to gain some insights into quantum gravity since the thermodynamic property of black hole is essentially a quantum feature of gravity. On the other hand, it is possible through the thermodynamics of black holes with AdS asymptotic to study the thermodynamic properties and phase structure of a ceratin field theory because due to the AdS/CFT correspondence, some higher derivative curvature terms can be regarded as the corrections of large N expansion of dual field theory. Among the higher derivative gravity theories, the so-called Lovelock gravity [1] is quite special, whose Lagrangian consists of the dimensionally extended Euler densities