Let M be a finitely generated graded module over a Noetherian homogeneous ring R with local base ring (R0, m0). Then, the nth graded component HR+i (M)n of the ith local cohomology module of M with respect to the irrelevant ideal R+ of R is a finitely generated R0-module which vanishes for all n ≫ 0. In various situations we show that, for an m0-primary ideal q0 ⊆ R0, the multiplicity eq0 (HR+i (M)n) of HR+i (M)n) is antipolynomial in n of degree less than i. In particular we consider the following three cases: (a) i < g(M), where g(M) is the so-called cohomological finite length dimension of M; (b) i = g(M) (c) dim (R0) = 2. In cases (a) and (b) we express the degree and the leading coefficient of the representing polynomial in terms of local cohomological data of M (e.g. the sheaf induced by M) on Proj (R). We also show that the lengths of the graded components of various graded submodules of (HR+i (M) are antipolynomial of degree less than i and prove invariance results on these degrees. © 2004 Elsevier B.V. All rights reserved.