  In this paper we study the behavior of the Casimir energy of a "multi-cavity"
across the transition from the metallic to the superconducting phase of the
constituting plates. Our analysis is carried out in the framework of the
ARCHIMEDES experiment, aiming at measuring the interaction of the
electromagnetic vacuum energy with a gravitational field. For this purpose it
is foreseen to modulate the Casimir energy of a layered structure composing a
multi-cavity coupled system by inducing a transition from the metallic to the
superconducting phase. This implies a thorough study of the behavior of the
cavity, where normal metallic layers are alternated with superconducting
layers, across the transition.
  Our study finds that, because of the coupling between the cavities, mainly
mediated by the transverse magnetic modes of the radiation field, the variation
of energy across the transition can be very large.
