Summary: This article deals with the study of the following critical exponent problem \[ (P_{\lambda, \mu}) \begin{cases} -\Delta_{\mathbb{H}^n} u+ (\mu g(x)-\lambda)u=u^{p-1} \text{ in } \mathbb{H}^n, \\ u>0 \text{ in } \mathbb{H}^n, \quad u\in H^1 (\mathbb{H}^n) \end{cases} \] where \(\mathbb{H}^n\) is the \(n\)-dimensional hyperbolic space, \(n \geq 4, p=2n/(n-2)\) is the critical exponent, \(\lambda, \mu >0\) are real parameters, \(\Delta_{\mathbb{H}^n}\) denotes the Laplace-Beltrami operator on \(\mathbb{H}^n\) and \(g(x)\) is a real valued potential function on \(\mathbb{H}^n\). Using variational methods, we establish the existence of positive ground state solution to \((P_{\lambda,\mu})\) and study the convergence of these solutions when \(\mu\) approaches \(+\infty\).