  In this paper we consider the production of top quark pairs close to
threshold and compute the dependence on the Higgs boson mass to order
$\alpha\alpha_s$. This requires the evaluation of the matching coefficient of
the vector current to two-loop level and the inclusion of Higgs mass dependent
operators in the non-relativistic effective theory. For Higgs boson masses
below 200 GeV moderate contributions to the top quark mass and the peak cross
section are observed. We also provide additional information to the on-shell
wave function renormalization which is relevant for the matching coefficient.
