The Hyperbolic Sombor index \( HSO(G) \) is a degree–based invariant obtained by assigning to each edge a weight that depends on the degrees of its end vertices and summing these contributions over the edge set. More precisely, \[ \mathrm{HSO}(G)=\sum_{v_1v_2\in E(G)}\frac{( d_G(v_1)^2+ d_G(v_2)^2)^{1/2}}{min\{d_G(v_1),d_G(v_2)\}}. \] where \( d_G(v_1) \) denotes the degree of vertex \( v_1 \) in the vertex set of \( G \). We give new bounds for Hyperbolic Sombor index and some well-known topological indices namely, geometric arithmetic index, symmetric division degree index, forgotten index, Albertson index, \( \sigma \)-irregularity index, diminished Sombor index.