  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 ($M_t$) effects. We present results for the total cross
section and Higgs pair invariant mass ($M_{hh}$) distribution. We obtain
results at next-to-leading logarithmic accuracy (NLL) which retain the full
$M_t$ dependence, and are matched to the full next-to-leading order (NLO)
prediction. Our NLL+NLO results represent the most advanced prediction with
full $M_t$ 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 $\mu_0 = M_{hh}/2$. We also consistently combine the full
NLL with the next-to-next-to-leading logarithmically (NNLL) accurate
resummation computed in the Born-improved large-$M_t$ limit, and match it to
the next-to-next-to-leading order approximation of Ref.
\cite{Grazzini:2018bsd}, so called NNLO_FTa. We find that the resummation
effects are very small at NNLL for $\mu_0 = M_{hh}/2$, 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 $\mu_0 = M_{hh}$, resulting in a more stable cross section with
respect to scale variations than the fixed order calculation.
