In this paper, we study a class of flows of closed, star-shaped hypersurfaces in hyperbolic space \(\mathbb {H}^{n+1}\) with speed \((\sinh r)^{{\alpha }/{\beta }} \sigma _{k}^{{1}/{\beta }}\) , where \(\sigma _{k}\) is the k-th elementary symmetric polynomial of the principal curvatures, \(\alpha \) , \( \beta \) are positive constants and r is the distance from points on the hypersurface to the origin. We obtain convergence results under some assumptions of k, \(\alpha \) and \( \beta \) . When \(k = 1, \alpha > 1 + \beta \) , and the initial hypersurface is mean convex, we prove that the mean convex solution to the flow for \( k=1 \) exists for all time and converges smoothly to a sphere. When \(1\le k \le n, \alpha > k+\beta \) , and the initial hypersurface is uniformly convex, we prove that the uniformly convex solution to the flow exists for all time and converges smoothly to a sphere. In particular, we generalize Li-Sheng-Wang’s results in [10] from Euclidean space to hyperbolic space.
