  Our purpose in this paper is to apply some maximum principles in order to
study the rigidity of complete spacelike hypersurfaces immersed in a spatially
weighted generalized Robertson-Walker (GRW) spacetime, which is supposed to
obey the so called strong null convergence condition. Under natural constraints
on the weight function and on the f-mean curvature, we establish sufficient
conditions to guarantee that such a hypersurface must be a slice of the ambient
space. In this setting, we also obtain new Calabi-Bernstein type results
concerning entire graphs in a spatially weighted GRW spacetime.
