It is well known that Error Bound conditions provide some (usually linear or sublinear) rate of convergence for gradient descent methods in unconstrained and, in certain cases, in constrained optimization. We prove that the same conditions in constrained optimization guarantee stability of minimization problems: if we slightly change the function and the set then the solution set can not change much. Both the function and the set are not necessarily convex. We obtain an upper bound for the Hausdorff halfdistance between solutions via the function from the Error bound condition. In a real Hilbert space or in Rn these results generalize known results about stability of convex functionals.