Note on Von Neumann Entropy and the Ordering of Inverse Temperatures

I show that for two inverse temperatures ${\beta }_1$ and ${\beta }_2$, the von Neumann entropy $S({\rho }_{\beta })$ of the Gibbs state ${\rho }_{\beta }$ for a given Hamiltonian $H$ satisfies $S({\rho }_{{\beta }_1})\ge S({\rho }_{{\beta }_2})⟺{\beta }_1\le {\beta }_2$. That is, von Neumann entropy is a monotonically increasing function of temperature.