For conformally related Einstein spacetimes, equilibrium fluid temperature and chemical potential both scale as the inverse conformal factor, preserving the ratio μ/T.
How the Schwarzschild-de Sitter horizons remain in thermal equilibrium at vastly different temperatures
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abstract
The Tolman-Ehrenfest criterion of thermal equilibrium for a static fluid in a static spacetime is generalized to stationary heat conduction, in the approximation in which backreaction is negligible. Applying this generalized criterion to the Hawking radiation in the Schwarzschild-de Sitter geometry shows that the two horizons (which act as thermostats) remain in thermal equilibrium. The temperature of the radiation fluid interpolates between the temperatures at the horizons, with a static analytic profile that is given explicitly.
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Thermodynamic parameters of fluids on conformally connected spacetimes
For conformally related Einstein spacetimes, equilibrium fluid temperature and chemical potential both scale as the inverse conformal factor, preserving the ratio μ/T.