  Compared with twisted bilayer graphene, twisted double bilayer graphene
(TDBG) provides another important platform to realize the moir\'e flat bands.
In this paper, we first calculate the valley Chern number phase diagram of TDBG
in the parameter space spanned by the twist angle and the interlayer electric
potential. To include the effects of interactions, we then phenomenologically
introduce the spin-splitting and valley-splitting. We find that when the valley
splitting is larger than the bandwidth of the first conduction band so that a
gap is opened and the spin splitting is relatively weak, the orbital Chern
insulator emerges at half-filling, associated with a large orbital
magnetization (OM). Further calculations suggest that there is no sign reversal
of the OM when the Fermi energy goes from the bottom to the top of the
half-filling gap, as the OM remains negative in both AB-AB stacking and AB-BA
stacking. The implications of our results for the ongoing experiments are also
discussed.
