  In this paper, we study the Casimir effect in a curved spacetime described by
gravitational actions quadratic in the curvature. In particular, we consider
the dynamics of a massless scalar field confined between two nearby plates and
compute the corresponding mean vacuum energy density and pressure in the
framework of quadratic theories of gravity. Since we are interested in the
weak-field limit, as far as the gravitational sector is concerned we work in
the linear regime. Remarkably, corrections to the flat spacetime result due to
extended models of gravity (although very small) may appear at the first-order
of our perturbative analysis, whereas general relativity contributions start
appearing at the second order. Future experiments on the Casimir effect might
represent a useful tool to test and constrain extended theories of gravity.
