For an integer $k\geq 1$, the objective of \textsc{$k$-Geodesic Center} is to find a set $\mathcal{C}$ of $k$ isometric paths such that the maximum distance between any vertex $v$ and $\mathcal{C}$ is minimised. Introduced by Gromov, \emph{$δ$-hyperbolicity} measures how treelike a graph is from a metric point of view. Our main contribution in this paper is to provide an additive $O(δ)$-approximation algorithm for \textsc{$k$-Geodesic Center} on $δ$-hyperbolic graphs. On the way, we define a coarse version of the pairing property introduced by Gerstel \& Zaks (Networks, 1994) and show it holds for $δ$-hyperbolic graphs. This result allows to reduce the \textsc{$k$-Geodesic Center} problem to its rooted counterpart, a main idea behind our algorithm. We also adapt a technique of Dragan \& Leitert, (TCS, 2017) to show that for every $k\geq 1$, $k$-\textsc{Geodesic Center} is NP-hard even on partial grids.