  Given a Gaussian analytic function $f_L$ of intesity $L$ in the unit ball of
$\mathbb C^n$, $n\geq 2$, consider its (random) zero variety $Z(f_L)$. We study
the variance of the $(n-1)$-dimensional volume of $Z(f_L)$ inside a
pseudo-hyperbolic ball of radius $r$. We first express this variance as an
integral of a positive function in the unit disk. Then we study its asymptotic
behaviour as $L\to\infty$ and as $r\to 1^{-}$. Both the results and the proofs
generalise to the ball those given by Jeremiah Buckley for the unit disk.
