  We study functions on the class group of a toric variety measuring the rates
of growth of the cohomology groups of multiples of divisors. We show that these
functions are piecewise polynomial with respect to finite polyhedral chamber
decompositions. As applications, we express the self-intersection number of a
T-Cartier divisor as a linear combination of the volumes of the bounded regions
in the corresponding hyperplane arrangement and prove an asymptotic converse to
Serre vanishing.
