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The cosmological event horizon found in a de Sitter universe has some interesting similarities to that of a black hole. For example, since we can find a temperature at the horizon, we are able to use the relation $S = A/4G$ to calculate the entropy of the horizon related to it's surface.

I am looking for a method of directly calculating the entropy, instead of using the famous relation given above. One example I found was mentioned in Les Houches Lectures on De Sitter Space (pages 17,18). Unfortunately, the method is given as an exercise, and the conclusion given is far from obvious (to me).

They start by looking at the static Schwarzschild-de Sitter metric in three dimensions given by $$\mathrm{d}s^2 = -(1 - 8GE - r^2)\mathrm{d}t^2 + \dfrac{\mathrm{d}r^2}{1-8GE-r^2} + r^2 \mathrm{d}\theta^2 $$

The paper states that finding a Green function for $SdS_3$ by analytic continuation from the smooth Euclidian solution periodic in $\tau \rightarrow \tau + \dfrac{2 \pi i}{\sqrt{1 - 8GE}}$ will give all required answers to find the entropy (the author continues summing up temperature, invoking a thermodynamic identity and finishes off with $S$.

I have yet to find another paper that uses this derivation, but I am quite interested in the part missing. Is there someone who can shed a light on how finding such a Green function leads to temperature/entropy?

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