  In van der Waals multilayers of triangular lattice, trigonal warping occurs
universally due to the interlayer hopping. We theoretically investigate the
effect of trigonal warping upon distinctive topological phases, like the
quantum anomalous Hall effect (QAHE) and the quantum valley Hall effect (QVHE).
Taking Bernal-stacked bilayer graphene as an example, we find that the trigonal
warping plays a crucial role in the formation of QAHE in large exchange field
and/or interlayer potential difference by inducing extra band inversion points
at momentum further away from high symmetric point. The presence of trigonal
warping shrinks the phase space of QAHE and QVHE, leading to the emergence of
valley-polarized QAHE with high Chern numbers ranging from $ \mathcal{C}=-7 $
to $ 7 $. These results suggest that the universal trigonal warping may play
important role when the Bloch states at momentum away from high-symmetric
points are involved.
