$θ$-almost twisted Poisson cohomology

We introduce the notion of a $θ$-almost twisted Poisson structure on manifolds, which involves incorporating a closed $1$-form $θ$ into twisted Poisson structures under specific conditions. We provide a characterization of this structure on low-dimensional manifolds and construct the Lie-Rinehart algebra on the module of $1$-forms on manifolds equipped with this structure. This construction leads to a cochain complex and its associated cohomology, which we refer to as $θ$-almost twisted Poisson cohomology. An example illustrating this cohomology is also presented on $\mathbb{R}^5$.