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.