Thermal aspects of the anomalous $\omega \to \mathrm{\infty }$ limit of Brans-Dicke gravity

Brans-Dicke gravity does not always reduce to General Relativity in the limit $\omega \to \mathrm{\infty }$ for the coupling constant. This anomalous behavior is examined within the formalism of the first-order thermodynamics of scalar-tensor gravity. It is shown that this effect is linked to the non-vanishing nature of the chemical potential, in the Einstein frame formulation of the thermodynamic analogy, in the $\omega \to \mathrm{\infty }$ limit.