Thermodynamics of a quantum corrected Schwarzschild black hole in AdS spacetime

The thermodynamics for a quantum corrected Schwarzschild black hole with AdS background is investigated. First, the black hole metric is introduced and given when the spherical symmetric quantum fluctuations of spacetime are considered for Schwarzschild AdS black hole. Then, based on AdS background, we get the free energy for the quantum corrected black hole by AdS gravity renormalization and saddle-point approximation. By the spherical symmetric quantum fluctuation in AdS spacetime is considered, there are three main results found. Firstly, due to quantum fluctuation, there is a divergence term in the Euclidean action, which contributes to the energy of spacetime. Secondly, the entropy of the modified black hole is still proportional to the horizon area. But when the contribution of quantum fluctuations is deducted, the relationship between entropy and area is not so. It shows that, the area entropy theorem for the black hole is the both effect of quantum fluctuations and black hole. Thirdly, quantum fluctuation doesn't affect the temperature of Hawking-Page transition, but the radius of the black hole of Hawking-Page transition will be corrected.

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