  An important task of the LHC is the investigation of the Higgs-boson sector.
Of particular interest is the reconstruction of the Higgs potential, i.e. the
measurement of the Higgs self-couplings. Based on previous analyses, within the
2HDMs type~I and~II, we analyze several two-dimensional benchmark planes that
are over large parts in agreement with all theoretical and experimental
constraints. For these planes we evaluate di-Higgs production cross sections at
the (HL-)LHC with a center-of-mass energy of 13 TeV at NLO in the heavy
top-quark limit with the code HPAIR. We investige in particular the process $gg
\to hh$, with $h$ being the Higgs boson discovered at the LHC with a mass of
about 125 GeV. The top box diagram of the loop-mediated gluon fusion process
into Higgs pairs interferes with the $s$-channel exchange of the two CP-even
2HDM Higgs bosons $h$ and $H$ involving the trilinear couplings $\lambda_{hhh}$
and $\lambda_{hhH}$, respectively. Depending on the size of the involved
top-Yukawa and triple Higgs couplings as well as on the mass of $H$, the
contribution of the $s$-channel $H$~diagram can be dominating or be highly
suppressed. We find regions of the allowed parameter space in which the
di-Higgs production cross section can differ by many standard deviations from
its SM prediction, indicating possible access to deviations in $\lambda_{hhh}$
from the SM value $\lambda_{\rm SM}$ and/or contributions involving
$\lambda_{hhH}$. The sensitivity to $\lambda_{hhH}$ is further analyzed
employing the $m_{hh}$ distributions. We demonstrate how a possible measurement
of $\lambda_{hhH}$ depends on the various experimenal uncertainties. Depending
on the underlying parameter space, the HL-LHC may have the option not only to
detect beyond-the-Standard-Model triple Higgs couplings, but also to provide a
first rough measurement of their sizes.
