We consider the family $\{f_L\}_{L>0}$ of Gaussian analytic functions in the unit disk, distinguished by the invariance of their zero set with respect to hyperbolic isometries. Let $n_L\left(r\right)$ be the number of zeros of $f_L$ in a disk of radius $r$. We study the asymptotic probability of the rare event where there is an overcrowding of the zeros as $r\uparrow1$, i.e. for every $L>0$, we are looking for the asymptotics of the probability $\mathbb{P}\left[n_L(r)\geq V(r)\right]$ with $V\left(r\right)$ large compared to the $\mathbb{E}\left[n_L\left(r\right)\right]$. Peres and Virág showed that for $L=1$ (and only then) the zero set forms a determinantal point process, making many explicit computations possible. Curiously, contrary to the much better understood planar model, it appears that for $L<1$ the exponential order of decay of the probability of overcrowding when $V$ is close to $\mathbb{E}\left[n_L\left(r\right)\right]$ is much less than the probability of a deficit of zeros.