Quasiconcavity of the Variance of the Gibbs Measure

Pardon me for taking some time to set up my notation as I do not come from a physics background and am not using the standard statistical mechanics notation. Consider a Gibbs measure on a set of states $mathcal{X}$,
$$
p(x) = frac{exp(-beta H(x))}{Z_beta},
$$
where
$$
Z_beta = int_{xin mathcal{X}} exp(-beta H(x)) mathrm{d}x,
$$
is the partition function. The expectation and variance of $H(x)$ are then:

$$
mathbb{E}[H] = int_{xin mathcal{X}} H(x)frac{exp(-beta H(x))}{Z_beta} mathrm{d}x,
$$

$$
mathbb{V}[H] = mathbb{E}[H^2] – mathbb{E}[H]^2.
$$

These two quantities are related through by derivatives of the log-partition function: $mathbb{E}[H] = -frac{partial Z_beta}{partial beta}$, $mathbb{V}[H] = frac{partial^2 Z_beta}{partial beta^2}$.

The Question

It appears to me through numerical simulation that $mathbb{V}[H]$ is a quasiconcave function of $beta$. That is, it is roughly unimodal over $beta in (-infty, infty)$. I have done a large number of numerical simulations for the case where the set $mathcal{X}$ is discrete, and this property seems remarkably persistent. I have attached a graph demonstrating the shape of $mathbb{V}[H]$ showing this shape, although the peak of the function is not always as close to $beta = 0$ as depicted here.

Does this quasiconcave property hold generically? Is this a known result or easy to prove? If not, I am willing to make assumptions about the nature of $H(x)$ to achieve this result. More generally, I am interested in the quasiconcavity of $frac{mathbb{V}[H]}{mathbb{E}[H]}$, but understanding the quasiconcavity of the variance seems like an important first step here.