  We study the Casimir vacuum energy density and the Casimir pressure for a
massless scalar field confined between two nearby parallel plates in a slightly
curved, static spacetime background, employing the weak field approximation in
the framework of Extended Theories of Gravity (ETG). Following a perturbative
approach upto second order, we find the gravity correction in the ETG to
Casimir vacuum energy density and pressure. The corrections to the vacuum
energy density in presence of curved spacetime in the framework of General
Relativity (GR) are small and today they are still undetected with the current
technology. However, future sensitivity improvement in gravitational
interferometer experiments will give an useful tool to detect such effect
induced by gravity. For these reason we retain interesting from a theoretical
point of view generalize the outcomes of GR in the context of ETG. Finally, we
find the general relation to constraining the free parameters of the ETG.
