We establish new uniqueness and nonexistence results concerning spacelike hypersurfaces immersed with constant \(\varphi \) -mean curvature in a spatially weighted generalized Robertson–Walker (GRW) spacetime \(-I\times _fM^n_\varphi \) endowed with a weight function \(\varphi \) , via several maximum principles jointly with a suitable sharp inequality derived from the Bochner formula related to the drift Laplacian. For this, we assume that the ambient spacetime satisfies some curvature constraints, one of which can be regarded as a natural extension of the timelike convergence condition for the context of the Bakry-Émery-Ricci tensor. We also derive applications to the study of the constant \(\varphi \) -mean curvature spacelike hypersurface equation in spatially weighted GRW spacetimes, as well as to some standard models of GRW spacetimes, like the Einstein–de Sitter and anti-de Sitter spaces and Lorentzian products whose fiber is the Gaussian space.
