  We introduce a novel set of stability conditions for vacua with broken
Lorentz symmetry. The first class of conditions require that the energy be
minimized under small geometric deformations, which translates into requiring
the positivity of a "stiffness" four-tensor. The second class of conditions
requires that stress forces be restoring under small deformations. We then
apply these conditions to examples of recently-discovered spatially modulated
(or "striped") phases in holographic models of superconductors and high-density
QCD. For backreacted solutions we find that the pressure condition is
equivalent to thermodynamic stability. For probe solutions, however, these
conditions are in conflict with the minimization of the free energy. This
suggests that either the solutions are unstable or the definition of the free
energy in the probe approximation must be revised for these solutions.
