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.