Generalizing the continuous rank function of Barja-Pardini-Stoppino, in this paper
we consider cohomological rank functions of Q-twisted (complexes of) coherent sheaves on abelian
varieties. They satisfy a natural transformation formula with respect to the Fourier-Mukai-Poincar´e
transform, which has several consequences. In many concrete geometric contexts these functions
provide useful invariants. We illustrate this with two different applications, the first one to GVsubschemes and the second one to multiplication maps of global sections of ample line bundles on abelian varieties.