  We discuss production of $D^0 D^0$ (and ${\bar D}^0 {\bar D}^0$) pairs
related to the LHCb Collaboration results for $\sqrt{s}$ = 7 TeV in
proton-proton scattering. We consider double-parton scattering (DPS) mechanisms
of double $c \bar c$ production and subsequent $cc \to D^{0}D^{0}$
hadronization as well as double $g$ and mixed $g c\bar c $ production with $gg
\to D^{0}D^{0}$ and $gc \to D^{0}D^{0}$ hadronization calculated with the help
of the scale-dependent hadronization functions of Kniehl et al. Single-parton
scattering (SPS) mechanism of digluon production is also taken into account. We
compare our results with several correlation observables in azimuthal angle
$\varphi_{D^{0}D^{0}}$ between $D^{0}$ mesons or in dimeson invariant mass
$M_{D^{0}D^{0}}$. The inclusion of new mechanisms with $g \to D^{0}$
fragmentation leads to larger cross sections, than when including only DPS
mechanism with standard scale-independent $cc \to D^{0}D^{0}$ fragmentation
functions. Some consequences of the presence of the new mechanisms are
discussed. In particular a larger $\sigma_{eff}$ is needed to describe the LHCb
data. There is a signature that $\sigma_{eff}$ may depend on transverse
momentum of $c$ quarks and/or $\bar c$ antiquarks.
