Let $\mu$ be a compactly supported probability measure on a finite-dimensional euclidean space (for simplicity) $\mathbb E$, and suppose $\mu$ has density. For a random point $x \sim \mu$, consider the Boltzmann distribution at temperature $T$, on $N$ states, viz
$$p_j(x; T) = \frac{\exp(-\frac{1}{T}E_j(x))}{\sum_{l=1}^N\exp(-\frac{1}{T}E_l(x))},$$
where energies $E_j(x) = x^Ty_j + c_j$ is the energy at state $j$, and $y_1,\ldots,y_N \in \mathbb E$, $c_1,\ldots, c_N \in \mathbb R$ are fixed.
Question
What can be said about the expectation integral $$e_j^\mu(T) := \int p_j(x; T) d\mu(x) $$
Is it a known quantity / transform ?
Is there an efficient way to compute $e_j^\mu(T)$ which is "cheaper" (in the sense faster convergence / lower variance of
estimates) than just sampling $x_1,\ldots, x_M \sim \mu$ and computing the empirical mean $\frac{1}{M}\sum_{i=1}^M p_j(x_i;T)$ ?
For the last two questions, for convenience one may assume $T \rightarrow 0^+$ (low-temperature limit). In the high-temperature limit, it's not hard to show that $e_j^\mu(\infty) = N^{-1},\; \forall j=1,\ldots,N$.
