$\sqrt{2 KL(f || g)}$ interpretation? I have seen in some papers that instead of using the Kullback-Leibler divergence $KL(f || g)$ between two probability density functions, $f$ and $g$, they use
$$\sqrt{2 KL(f || g)}.$$
Is there any motivation for this specific transformation?

This is explained in the paper you reference. In section 3.1:

Here, the factor “2” is introduced for convenience and the square root deals with the power of two that is naturally associated with the KLD (see the simple example at the end of this subsection).

Look at A2 page 36, and then section 4.1, page 16 Behaviour near the base model where there is an expansion of the KL divergence showing the quadratic behaviour for small distance (there $\xi$).
