If g and h are Riemannian metrics on the same manifold, say both of volume 1, then it follows (I guess from Poincaré duality) that their volume forms dvol_g and dvol_h are cohomologous.
Question: is there some explicitly given (n-1)-form such that $$d\omega=dvol_g-dvol_h$$
Let $n=\dim M$. You can construct an $n$-form on $M\times\mathbb R$ such that pulling it back by the maps $x\mapsto(x,0)\in M\times\mathbb R$ and $x\mapsto(x,1)\in M\times\mathbb R$ gives you your volume forms $\nu_0$ and $\nu_1$. It follows that $\nu_0$ and $\nu_1$ are cohomologous, and most proofs of the homotopy invariance of de Rham cohomology (like that given in Bott-Tu, say) construct explicitly what you want.