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We study constant mean curvature spacelike hypersurfaces in generalized Robertson–Walker spacetimes \(\overline{M}= I \times _f F\) which are spatially parabolic (i.e. its fiber F is a (non-compact) complete Riemannian parabolic manifold) and satisfy the null convergence condition. In particular, we provide several rigidity results under appropriate mathematical and physical assumptions. We pay special attention to the case where the GRW spacetime is Einstein. As an application, some Calabi–Bernstein type results are given.

Complete spacelike hypersurfaces in generalized Robertson–Walker and the null convergence condition: Calabi–Bernstein problems