Monotonicity of an expectation of monotonous function with unknown closed-form.

Question

Suppose that we are given an arbitrary set $\mathcal X \subseteq \mathbb R^d$, a probability density function of multivariate Gaussian distribution with mean $\mu$ and covariance matrix $\sigma^2I$, f$(x;\mu,\sigma)$, and a function $g: \mathbb R^d \rightarrow \mathbb R$ that is monotonically increasing. Can we say anything about the monotonicity of the expectation as a function of the standard deviation, $e(\sigma)=\int_{\mathcal X} g(x)f(x;\mu, \sigma) dx$ (the integral may not have closed-form answer)? If no, what additional assumptions on the problem formulations should be made or on which domain of $\sigma$ should be considered so that the monotonicity can be determined? If yes, how about $e(\sigma)=\int_{\mathcal X} g\circ h(x)f(x;\mu, \sigma) dx$ for any $h: \mathbb R^d \rightarrow \mathbb R$ and monotonically increasing $g: \mathbb R \rightarrow \mathbb R$?