Consider two probability measures $\mu_0$ and $\mu_1$ on $\mathbb{R}^n$, such that $\mu_0\ll \mu_1$. Then I found a "distance" like quantitiy
$$
\mathrm{Var}_{\mu_1}\left(\frac{\mathrm{d}\mu_0}{\mathrm{d}\mu_1}\right) 
$$
Is this quantity already known?

For simplicity assume that both measures are absolute continuous with respect to the Lebesgue measure and we denote by $p_0$ and $p_1$ the densities. Hence the condition $\mu_0\ll \mu_1$ states that $\mathrm{supp}(p_0)\subseteq \mathrm{supp}(p_1)$. And the quantity above can be bounded below by the Kullback-Leibner divergence $K(p_0|p_1)$, just by using the linearization of the logarithm $\log x \leq x-1$. 
$$
 K(p_0| p_1) = \int p_0 \log \frac{p_0}{p_1} dx \leq \int \left(\frac{p_0^2}{p_1}-p_0\right)dx = \int \frac{p_0^2}{p_1^2}d\mu_1 - 1 = \mathrm{Var}_{\mu_1}\left(\frac{\mathrm{d}\mu_0}{\mathrm{d}\mu_1}\right) 
$$
I'm especially interested in conditions on the distributions for which this quantity becomes $+\infty$, are there some simple characterizations?

For instance, if one considers two Gaussians with equal mean and different variances, hence $\mu_0 = \mathcal{N}(0,1)$ and $\mu_1 = \mathcal{N}(0,\sigma^2)$, then
$$
  \mathrm{Var}_{\mu_1}\left(\frac{\mathrm{d}\mu_0}{\mathrm{d}\mu_1}\right) = \begin{cases}
   \frac{1}{\sigma\sqrt{2-\sigma^2}} &, \sigma^2 < 2 \\
   +\infty &,\sigma^2 \geq 2 . 
 \end{cases}
$$