Differential mass formula for de Sitter metric

I'm trying to write down a differential mass formula for the de Sitter metric, analogously to how Bardeen, Carter and Hawking did it in "The Four Laws of Black Hole Mechanics" but I'm running into some issues.

Consider the identity $$\nabla_b\nabla^b\xi^a=R^a_b\xi^b.$$ If we consider the timelike Killing vector $$\xi=\partial_t$$ we can calculate the mass. Integrating this expression over the spacelike region enclosed by the cosmological horizon $$S$$ (because $$\xi$$ becomes spacelike beyond the horizon) and using Stokes' theorem to get a surface integral on the left gives: $$\int_C\nabla^b\xi^ad\Sigma_{ab}=\int_SR^a_b\xi^bd\Sigma_a$$ where $$C$$ is the closed border of $$S$$ that coincides with the cosmological horizon. We can then use $$R_{ab}=8\pi(T_{ab}-\frac{1}{2}Tg_{ab})+\Lambda g_{ab}$$ to rewrite the right hand side, multiplying by $$(4\pi)^{-1}$$ then gives $$(4\pi)^{-1}\int_C\nabla^b\xi^ad\Sigma_{ab}=\int_S2(T_{ab}-\frac{1}{2}Tg_{ab})\xi^bd\Sigma^a +(4\pi)^{-1}\int_S\Lambda\xi^ad\Sigma_a$$ Here, the left hand side can be interpreted as the mass contained inside the cosmological horizon ($$M_C$$). The first term on the right hand side is the contribution of the matter inside the cosmological horizon and the second term is the mass due to the cosmological constant in the region enclosed by the cosmological horizon.

Using the definition of the surface gravity in asymptotically de Sitter space we can simplify the expression on the left; $$(4\pi)^{-1}\int_C\nabla^b\xi^ad\Sigma_{ab}=-\kappa_c A_c(4\pi)^{-1}.$$ Our first result is the mass of empty de Sitter space: $$M_C=-\kappa_c A_c(4\pi)^{-1}.$$ Now I would like to calculate the differential of this equation because I'm eventually interested in the thermodynamics of de Sitter. Following Bardeen, Carter and Hawking we choose a gauge in which $$\delta\xi^{a}=0$$. However, I have no clue on how to evaluate all the differential expressions and I hope someone can help me with that. The result is most likely $$\delta M_C=-\kappa_c\delta A_c(8\pi)^{-1}$$ but I don't know how to get there in the de Sitter case.

• What parameter are you varying for the differential? Cosmological constant? A related (potentially duplicate) question is physics.stackexchange.com/q/507491 but it does not have any answers. – A.V.S. Dec 12 '19 at 15:34