We perform the all orders resummation of threshold enhanced contributions for the Higgs boson pair production cross section via gluon fusion, including finite top quark mass (Mt) effects. We present results for the total cross section and Higgs pair invariant mass (Mhh) distribution. We obtain results at next-to-leading logarithmic accuracy (NLL) which retain the full Mt dependence, and are matched to the full next-to-leading order (NLO) prediction. Our NLL+NLO results represent the most advanced prediction with full Mt dependence for this process, and produce an increase of about 4% in the total cross section with respect to the NLO result for LHC energies, and for a central scale μ0 = Mhh/2. We also consistently combine the full NLL with the next-to-next-to-leading logarithmically (NNLL) accurate resummation computed in the Born-improved large-Mt limit, and match it to the next-to-next-to-leading order approximation of ref. , so called NNLOFTa. We find that the resummation effects are very small at NNLL for μ0 = Mhh/2, in particular below 1% at 13 TeV, indicating that the perturbative expansion is under control. In all cases the resummation effects are found to be substantially larger for the central scale μ0 = Mhh, resulting in a more stable cross section with respect to scale variations than the fixed order calculation.