We prove that a proper weak solution \(\{ \Omega _{t} \}_{0 \le t < \infty }\) to inverse mean curvature flow in hyperbolic space \(\mathbb {H}^{n}\), \(3 \le n \le 7\), is eventually smooth and star-shaped for an arbitrary initial domain \(\Omega _{0}\). In fact, this happens by the time \(\begin{aligned} T= (n-1) \log \left( \frac{\text {sinh} \left( r_{+} \right) }{ \text {sinh} \left( r_{-} \right) } \right) , \end{aligned}\)where \(r_{+}\) and \(r_{-}\) are the geodesic out-radius and in-radius of \(\Omega _{0}\). The approach is based on an Alexandrov reflection method for extrinsic curvature flows originally introduced by Chow-Gulliver [9]. In addition, our methods characterize expanding spheres as proper weak IMCF on \(\mathbb {H}^{n} \setminus \{ 0 \}\) for arbitrary n, thereby implying a result for ancient smooth solutions. As applications of the regularity theorem, we derive optimal Minkowski inequalities for arbitrary smooth domains of \(\mathbb {H}^{n}\), \(3 \le n \le 7\), extending those of Brendle-Hung-Wang [5] and De Lima-Girao [11]. From this, we also extend the Riemannian Penrose inequality from [11] to balanced asymptotically graphs over the exteriors of outer-minimizing domains in \(\mathbb {H}^{n}\).
