  Twisted Lie algebroid cohomologies, i.e. with values in representations, are
shown to be Lie algebroid homotopy-invariant. Several important classes of
examples are discussed. As an application, a generalized version of the
Poincar\'e lemma is given in the transitive case. Together with the Mayer-
Vietoris theorem, which holds in this more general context as well, this leads
to a K\"unneth formula for the cohomologies of Lie algebroids with values in
representations.
