Consider a fibration $F \to E \to B $, where $B $ is path connected. My question is can we use the cohomology Leray-Serre spectral sequence in case fiber is not connected? For example, when fiber is disjoint union of two spaces? In this case, does there exist a spectral sequence $E^{**}_r$ converge as an algebra to $H^*(E;R)$, with $$ E^{p,q}_2\cong H^p(B;\mathcal H^q(F;R)),$$ the cohomology of the space $B$ with local coefficient in the cohomology of the fiber.
Thank you so much in advance.