higher moments of entropy... does the variance of $\log x$ have any operational meaning?

The Shannon entropy is the average of the negative log of a list of probabilities $$\{ x_1 , \dots , x_d\}$$, i.e. $$H(x)= -\sum\limits_{i=1}^d x_i \log x_i$$ there are of course lots of nice interpretations of the Shannon entropy. What about the variance of $$-\log x_i$$ ? $$\sigma^2 (-\log x)=\sum\limits_i x_i (\log x_i )^2-\left( \sum\limits_i x_i \log x_i \right)^2$$ does this have any meaning / has it been used in the literature?

$\log 1/x_i$ is sometimes known as the 'surprise' (e.g. in units of bits) of drawing the symbol $x_i$, and $\log 1/X$, being a random variable, has all the operational meanings that come with any random variable, namely, entropy is the average 'surprise'; similarly, higher moments are simply higher moments of the surprise measure of $X$.