  We show that the valley Chern number of the low energy band in twisted double
bilayer graphene can be tuned through two successive topological transitions,
where the direct bandgap closes, by changing the electric field perpendicular
to the plane of the graphene layers. The two transitions with Chern number
changes of -3 and +1 can be explained by the formation of three satellite Dirac
points around the central Dirac cone in the moir\'e Brillouin zone due to the
presence of trigonal warping. The satellite cones have opposite chirality to
the central Dirac cone. Considering the overlap of the bands in energy, which
lead to metallic states, we construct the experimentally observable phase
diagram of the system in terms of the indirect bandgap and the anomalous valley
Hall conductivity. We show that while most of the intermediate phase becomes
metallic, there is a narrow parameter regime where the transition through three
insulating phases with different quantized valley Hall conductivity can be
seen. We systematically study the effects of variations in the model parameters
on the phase diagram of the system to reveal the importance of particle-hole
asymmetry and trigonal warping in constructing the phase diagram. We also study
the effect of changes in interlayer tunneling on this phase diagram.
