The author presents a simple model of a massless scalar field confined in cavity at the background of the static spacetime with a small curvature. Within the weak-field approximation he solves the Klein-Gordon equation up to second order of perturbation expansion. The quantized model after \(\zeta\)-function regularization gives the gravitationally induced corrections to the flat spacetime Casimir energy. The corrections are caused by the non-uniformity of the gravitational potential. The effect is too small to be detected using current experimental technology but it is interesting theoretically, as it might play a role in the cosmological scenario. There might also be expected, at microscopic level, the changes in vacuum fluctuations. The paper consists of seven sections.