For discrete probability distributions $P$ and $Q$ defined on the same sample space, $\mathcal{X}$, the Kullback-Leibler divergence is defined as
$$
D_{\mathrm{KL}}(P \parallel Q)=\sum_{x \in \mathcal{X}} P(x) \log \frac{P(x)}{Q(x)}
$$

Is there a notion of a higher Kullback-Leibler divergence? In other words, has the following divergence been studied:

$$
D^k_{\mathrm{KL}}(P \parallel Q)=\sum_{x \in \mathcal{X}} P(x) \left(\log \frac{P(x)}{Q(x)}\right)^k.
$$

This is with a view to generalizing Stam's inequality.