The paper gives the index growth for Delaunay unduloids, i.e. hypersurfaces of revolution with constant mean curvature \((cmc) 1\) in \(\mathbb{R}^n\). An explicit asymptotic index growth rate for \(cmc 1\) surfaces of finite topology in \(\mathbb{R}^3\) with properly embedded ends is obtained here by using results of N. J. Korevaar, R. Kusner and B. Solomon [J. Differ. Geom. 30, No. 2, 465-503 (1989; Zbl 0726.53007)]. Similar results are obtained for hypersurfaces with \(cmc>1\) in hyperbolic space.