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Almost all advanced GR textbooks will have the content of black hole thermodynamics for asymptotically flat black hole. And this paper solve the asymptotically AdS (Anti-de Sitter) black hole http://link.springer.com/article/10.1007/BF01208266. Why there is no black hole thermodynamics in asymptotically de-Sitter black hole?

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The reason is because you need a hypersurface similar to a Cauchy surface in order to be able to define thermodynamic quantities of the black hole like temperature, mass etc.

For example - if you want to calculate the mass of a black hole theoretically, you would first have to construct a global hypersurface at the asymptotic limit of spacetime and then evaluate the mass integral (or other thermodynamic quantities) over this hypersurface.

In de Sitter spacetime, this hypersurface lacks a well defined prescription. This is why thermodynamics in dS spacetime is quite difficult.

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1- Thermodynamics is certainly not related to existence of a Cauchy surface, because that surface doesn't exist for AdS where boundary conditions should also added, AdS is not globally hyperbolic.

2- Thermodynamics of de Sitter is defined, e.g. for Schwarzschild de Sitter, each horizon has a temperature but you can't define a temperature at infinity because this concept doesn't exist (because of the cosmological horizon).

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