  In this paper we discuss the stability of geodesic spheres in
$\mathbb{S}^{n+1}$ under constrained curvature flows. We prove that under some
standard assumptions on the speed and weight functions, the spheres are stable
under perturbations that preserve a volume type quantity. This extends results
by Escher and Simonett, 1998, and the author, 2015, to a Riemannian manifold
setting.
