  In this contribution, we study spacetimes of cosmological interest, without
making any symmetry assumptions. We prove a rigid Hawking singularity theorem
for positive cosmological constant, which sharpens known results. In
particular, it implies that any spacetime with $\operatorname{Ric} \geq -ng$ in
timelike directions and containing a compact Cauchy hypersurface with mean
curvature $H \geq n$ is timelike incomplete. We also study the properties of
cosmological time and volume functions, addressing questions such as: When do
they satisfy the regularity condition? When are the level sets Cauchy
hypersurfaces? What can one say about the mean curvature of the level sets?
This naturally leads to consideration of Hawking type singularity theorems for
Cauchy surfaces satisfying mean curvature inequalities in a certain weak sense.
