We present a class of exact analytic and static, spherically symmetric black hole solutions in the semi-classical Einstein equations with Weyl anomaly. The solutions have two branches, one is asymptotically flat and the other asymptotically de Sitter. We study thermodynamic properties of the black hole solutions and find that there exists a logarithmic correction to the well-known Bekenstein-Hawking area entropy. The logarithmic term might come from non-local terms in the effective action of gravity theories. The appearance of the logarithmic term in the gravity side is quite important in the sense that with this term one is able to compare black hole entropy up to the subleading order, in the gravity side and in the microscopic statistical interpretation side.

In this paper, by use of the holographic principle together with the equipartition law of energy and the Unruh temperature, we derive the Friedmann equations of a Friedmann-Robertson-Walker universe.

We find topological (charged) black holes whose horizon has an arbitrary constant scalar curvature $2k$ in Hořava-Lifshitz theory. Without loss of generality, one may take $k=1$, 0, and $-1$. The black hole solution is asymptotically anti–de Sitter with a nonstandard asymptotic behavior. Using the Hamiltonian approach, we define a finite mass associated with the solution. We discuss the thermodynamics of the topological black holes and find that the black hole entropy has a logarithmic term in addition to an area term. We find a duality in Hawking temperature between topological black holes in Hořava-Lifshitz theory and Einstein’s general relativity: the temperature behaviors of black holes with $k=1$, 0, and $-1$ in Hořava-Lifshitz theory are, respectively, dual to those of topological black holes with $k=-1$, 0, and 1 in Einstein’s general relativity. The topological black holes in Hořava-Lifshitz theory are thermodynamically stable.

We write a renormalization group equation for the function $f$ in a theory of gravity in the $f(R)$ truncation. Our equation differs from previous ones due to the exponential parametrization of the quantum fluctuations and to the choice of gauge. The cutoff procedure depends on three free parameters, and we find that there exist discrete special choices of parameters for which the flow equation has fixed points where $f=f_0+f_{1}R+f_2{R}^2$. For other values of the parameters the solution seems to be continuously deformed.

Lovelock gravity is a fascinating extension of general relativity, whose action consists of dimensionally extended Euler densities. Compared to other higher order derivative gravity theories, Lovelock gravity is attractive since it has a lot of remarkable features such as the fact that there are no more than second order derivatives with respect to the metric in its equations of motion, and that the theory is free of ghosts. Recently, in the study of black strings and black branes in Lovelock gravity, a special class of Lovelock gravity is considered, which is named pure Lovelock gravity, where only one Euler density term exists. In this paper we study black hole solutions in the special class of Lovelock gravity and associated thermodynamic properties. Some interesting features are found, which are quite different from the corresponding ones in general relativity.

We calculate the shear viscosity of field theories with gravity duals of Gauss-Bonnet gravity with a nontrivial dilaton using anti-de Sitter/conformal field theory. We find that the dilaton field has a nontrivial contribution to the ratio of shear viscosity over entropy density, and, after imposing a causal constraint for the boundary field theory, the new lower bound $4/25\pi$, obtained from pure Gauss-Bonnet gravity, may have a small violation.