  We study how the cohomology of a type $F_\infty$ relatively hyperbolic group
pair $(G,\mathcal{P})$ changes under Dehn fillings (i.e. quotients of group
pairs). For sufficiently long Dehn fillings where the quotient pair
$(\bar{G},\bar{\mathcal{P}})$ is of type $F_\infty$, we show that there is a
spectral sequence relating the cohomology groups $H^i(G,\mathcal{P};\mathbb{Z}
G)$ and $H^i\left(\bar{G},\bar{\mathcal{P}};\mathbb{Z}\bar{G}\right)$. As a
consequence, we show that essential cohomological dimension does not increase
under these Dehn fillings.
