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 $S(\rho_{\beta_1}) \geq S(\rho_{\beta_2}) \iff \beta_{1} \leq \beta_{2}$. That is, von Neumann entropy is a monotonically increasing function of temperature.