  We study inclusive double-Higgs boson production at the LHC and at the HL-LHC
including variations of the trilinear and of the quartic Higgs boson
self-couplings at next-to-leading order (NLO) in QCD with full top quark mass
dependence. Our results include the two-loop contributions to the $gg
\rightarrow HH$ amplitudes that involve a modified $h_4$ vertex calculated in
arXiv:1810.04665. We present results at 13, 13.6 and 14 TeV centre-of-mass
energies. The implementation of the calculation is made publicly available in
the POWHEG-BOX-V2 Monte Carlo framework.
