  Let $n$ be a large integer, and let $G$ be the standard Gaussian vector in
$R^n$. Paouris, Valettas and Zinn (2015) showed that for all $p\in[1,c\log n]$,
the variance of the $\ell_p^n$--norm of $G$ is equivalent, up to a constant
multiple, to $\frac{2^p}{p}n^{2/p-1}$, and for $p\in[C\log n,\infty]$,
$\mathbb{Var}\|G\|_p\simeq (\log n)^{-1}$. Here, $C,c>0$ are universal
constants. That result left open the question of estimating the variance for
$p$ logarithmic in $n$. In this note, we resolve the question by providing a
complete characterization of $\mathbb{Var}\|G\|_p$ for all $p$. We show that
there exist two transition points (windows) in which behavior of
$\mathbb{Var}\|G\|_p$, viewed as a function of $p$, significantly changes. We
also discuss some implications of our result in context of random Dvoretzky's
theorem for $\ell_p^n$.
