  Let $e(s)$ be the error term of the hyperbolic circle problem, and denote by
$e_\alpha(s)$ the fractional integral to order $\alpha$ of $e(s)$. We prove
that for any small $\alpha>0$ the asymptotic variance of $e_\alpha(s)$ is
finite, and given by an explicit expression. Moreover, we prove that
$e_\alpha(s)$ has a limiting distribution.
