dg.differential geometry: a cohomology associated with a vector field in a variety of Riemann

Leave $ X $ be a vector field in a Riemannian variety $ (M, g) $. So we have a $ 1 $-to form $ beta $ with $ beta (Y) =_g $.

We consider the following subcomplex of the Rham complex $ Omega ^ * (M) $: $$ { alpha en Omega ^ * (M) | d alpha = alpha wedge beta } $$

This generates a cohomology with the standard external derivation. For the zero vector field we obtain the standard Rham cohomology.

Is this cohomology always a finite dimensional space? Does it depend on the Riemannian structure? Does it contain any dynamic information about the vector field? $ X $?

Is there an appropriate analogy of this cohomology in algebraic topology or another type of cohomologist? (In the context of algebraic topology, we replace $ 1 $-to form $ beta $ with a $ 1 $-cochain and the outer derivative with cup product)