Let $F \rightarrow E \rightarrow B$ be a fibre bundle such that $B$ is a smooth and compact manifold and $F$ obtains an associative H-space structure. Explicitly, it is not a principal bundle.

One may use the Leray-Serre spectral sequence to compute the (co)homology of $E$. But it does not make use of the H-space structure of $F$.

Is there a spectral sequence or some other tool which one can use in order to compute the cohomology or homology of $E$ by using the H-space structure of $F$?