Temperature in Friedmann thermodynamics and its generalization to arbitrary space-times

In recent articles we have introduced Friedmann thermodynamics, where certain geometric parameters in Friedmann models were treated like their thermodynamic counterparts (temperature, entropy, Gibbs potential, etc.). This model has the advantage of allowing us to determine the geometry of the universe by thermodynamic stability arguments. In this paper, in search for evidence for the definition of “gravitational” temperature, we will investigate a massless conformal scalar field in an Einstein universe in detail. We will argue that the “gravitational” temperature of the Einstein universe is given asT g=1/2π) (ħc/k) (1/R 0), where R0 is the radius of the Einstein universe. This is in accord with our definition of “gravitational” temperature in Friedmann thermodynamics and determines the dimensionless constant as 1/2π. We discuss the limitations of the model we are using. We also suggest a method to generalize our “gravitational” temperature to arbitrary space-times granted that they are sufficiently smooth