  We discuss production of two pairs of $c \bar c$ in proton-proton collisions
at the LHC. Both double-parton scattering (DPS) and single-parton scattering
(SPS) contributions are included in the analysis. Each step of DPS is
calculated within $k_t$-factorization approach. The conditions how to identify
the DPS contribution are presented. The discussed mechanism leads to the
production of pairs of mesons: each containing $c$ quarks or each containing
$\bar c$ antiquarks. We discuss corresponding production rates and some
differential distributions for $(D^0 D^0$ + $\bar D^0 \bar D^0)$ production.
Within large theoretical uncertainties the predicted DPS cross section is
fairly similar to the cross section measured recently by the LHCb
collaboration. The best description is obtained with the Kimber-Martin-Ryskin
(KMR) unintegrated gluon distribution. The contribution of SPS, calculated in
the high-energy approximation, turned out to be rather small. Finally, we
emphasize significant contribution of DPS mechanism to inclusive charmed meson
spectra measured recently by ALICE, ATLAS and LHCb.
