  We investigate the entanglement entropy of a massive scalar field
nonminimally coupled to spacetime curvature, assuming a static, spherically
symmetric background. We discretize the field Hamiltonian by introducing a
lattice of spherical shells and imposing a cutoff in the radial direction. We
then study the ground state of the field and quantify deviations from area law
due to nonminimal coupling, focusing in particular on Schwarzschild-de Sitter
and Hayward spacetimes, also discussing de Sitter spacetime as a limiting case.
We show that large positive coupling constants can significantly alter the
entropy scaling with respect to the boundary area, even in case of small field
mass. Our outcomes are interpreted in view of black hole entropy production and
early universe scenarios.
