This talk is about a quantitative stability result for geodesic spheres in rank-one symmetric spaces of non-compact type — including real, complex, quaternionic, and octonionic hyperbolic spaces. These spaces have negatively pinched sectional curvature, whose minima is distributed according to the underlying algebraic structure. This geometric framework, and in particular the distribution on the tangent space associated with any radial vector fields, plays a central role in the analysis. We show that geodesic spheres are uniformly stable under small volume-preserving $C^1$-perturbations, with perimeter gain controlled by the $W^{1,2}$-norm of the perturbation. As a consequence, we give a quantitative proof that, for small volumes, geodesic spheres are the unique isoperimetric regions.