King-Plosser-Rebello Preferences: Scale leisure KPR preferences are given by

$$ U(c, l) = \frac{\left(cv(l)\right)^{1-\sigma}-1}{1-\sigma}$$

with concave increasing $v$ and $c$, $l$ denoting consumption and leisure. In the limiting case of $\sigma\to 1$, we receive the standard additively-separable preferences 

$$ U(c,l) = \log c + v(l)$$

In the latter case, if I want to scale the relevance of leisure in total preferences, I can rewrite the preferences as $\log c + Av(l)$, and use $A$ for this matter.

How do I do that - scale leisure - in the general case?

If I rewrite the nominator as $(c A v(l))^{1-\sigma}$, it is not clear to me whether $A$ is scaling $c$ or $l$ (probably neither).
If you specify

$$ U(c, l) =\left(c\cdot [(v(l))^A]\right)^{1-\sigma}$$

then 

$$\ln U(c,l) = (1-\sigma)\ln c + A(1-\sigma)\ln[v(l)]$$

...and $A$ can be seen as regulating the relative weight of leisure, for any $v(l)$.