  We consider cusped hyperbolic $n-$manifolds, and compute \v{C}ech cohomology
groups of the Morse boundaries of their fundamental groups. In particular, we
show that the reduced \v{C}ech cohomology with real coefficients vanishes in
dimension at most $n-3$ and does not vanish in dimension $n-2$. A similar
result holds for relatively hyperbolic groups with virtually nilpotent
peripherals and Bowditch boundary homeomorphic to a sphere; these include all
non-uniform lattices in rank$-1$ simple Lie groups.
