Why can only asymptotically flat and AdS black hole have the thermodynamics? What's about asymptoticaly dS black hole?

Question

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?

Answer 1

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.

Answer 2

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).