  We consider the Hardy constant associated with a domain in the
$n$-dimensional Euclidean space and we study its variation upon perturbation of
the domain. We prove a Fr\'{e}chet differentiability result and establish a
Hadamard-type formula for the corresponding derivatives. We also prove a
stability result for the minimizers of the Hardy quotient. Finally, we prove
stability estimates in terms of the Lebesgue measure of the symmetric
difference of domains.
