AdS Black Hole with Phantom Scalar Field

In this paper, we present an AdS black hole solution with Ricci flat horizon in Einstein-phantom scalar theory. The phantom scalar fields just depend on the transverse coordinates $x$ and $y$, and which are parameterized by the parameter $\alpha$. We study the thermodynamics of the AdS phantom black hole. Although its horizon is a Ricci flat Euclidean space, we find that the thermodynamical properties of the black hole solution are qualitatively same as those of AdS Schwarzschild black hole. Namely there exists a minimal temperature, the large black hole is thermodynamically stable , while the smaller one is unstable, so there is a so-called Hawking-Page phase transition between the large black hole and the thermal gas solution in the AdS spacetime in Poincare coordinates. We also calculate the entanglement entropy for a strip geometry dual to the AdS phantom black holes and find that the behavior of the entanglement entropy is qualitatively the same as that of the black hole thermodynamical entropy.