  We study the role of the threshold variable in soft-gluon threshold
resummation. We focus on the computation of the resummed total cross section,
the final-state invariant-mass distribution, and transverse-momentum
distribution of the Higgs boson when produced in association with a
top-anti-top quark pair for the Large Hadron Collider operating at 13 TeV. We
show that different choices for the threshold variable result in differences at
next-to-leading power, i.e. contributions that are down by one power of the
threshold variable. These contributions are noticeable numerically, although
their effect on the resummed observables lies within the scale uncertainty of
those observables. The average central results, obtained after combining
several central-scale choices, agree remarkably well for different choices of
the threshold variable. However, different threshold choices do effect the
resulting scale uncertainty. To compute our results, we introduce a novel
numerical method that we call the deformation method, which aids the
stabilization of the inverse Mellin transform in cases where the analytical
Mellin transform of the partonic cross section is unknown. We show that this
method leads to a factor of 10 less function evaluations, while gaining a
factor of 4-5 in numerical precision when compared to the standard method.
