  We study the cohomology of group theoretic Dehn fillings. Applying the
Cohen-Lyndon property for sufficiently deep Dehn fillings of hyperbolically
embedded subgroups $H\hookrightarrow_h G$, obtained by the second named author,
we derive a spectral sequence that computes the cohomology of the corresponding
Dehn filling quotients $\overline{G}$. As an application, we establish an
isomorphism between the relative cohomology of the group pair $(G, H)$ and its
sufficiently deep Dehn filling quotient pair $(\overline{G}, \overline{H})$.
This allows us to generalise the results of Fujiwara and Manning on simplicial
volume of Dehn fillings of hyperbolic manifolds to Dehn fillings of Poincar\'e
duality pairs.
  We also strengthen the results of Olshanskii, Dahmani-Guirardel-Osin and Hull
on SQ-universality and common quotients of acylindrically hyperbolic groups by
adding cohomological finiteness conditions. We apply these results to obtain
hyperbolic and acylindrically hyperbolic quotients with special properties.
