  Let $R$ be the face ring of a simplicial complex of dimension $d-1$ and
${\mathcal R}(\mathfrak{n})$ be the Rees algebra of the maximal homogeneous
ideal $\mathfrak{n}$ of $R.$ We show that the generalized Hilbert-Kunz function
$HK(s)=\ell({\mathcal R}(\mathfrak n)/(\mathfrak n, \mathfrak n t)^{[s]})$ is
given by a polynomial for all large $s.$ We calculate it in many examples and
also provide a Macaulay2 code for computing $HK(s).$
