  We study the topological properties of a Haldane model on a band deformed
dice lattice, which has three atoms per unit cell (call them as A, B and C) and
the spectrum comprises of three bands, including a flat band. The bands are
systematically deformed with an aim to study the evolution of topology and the
transport properties. The deformations are induced through hopping anisotropies
and are achieved in two distinct ways. In one of them, the hopping amplitudes
between the sites of B and C sublattices and those between A and B sublattices
are varied along a particular direction, and in the other, the hopping between
the sites of A and B sublattices are varied (keeping B-C hopping unaltered)
along the same direction. The first case retains some of the spectral features
of the familiar dice lattice and yields Chern insulating lobes in the phase
diagram with $C=\pm2$ till a certain critical deformation. The topological
features are supported by the presence of a pair of chiral edge modes at each
edge of a ribbon and the plateaus observed in the anomalous Hall conductivity
support the above scenario. Whereas, a selective tuning of only the A-B hopping
amplitudes distorts the flat band and has important ramifications on the
topological properties of the system. The insulating lobes in the phase diagram
have distinct features compared to the case above, and there are dips observed
in the Hall conductivity near the zero bias. The dip widens as the hopping
anisotropy is made larger, and thus the scenario registers significant
deviation from the familiar plateau structure observed in the anomalous Hall
conductivity. However, a phase transition from a topological to a trivial
insulating region demonstrated by the Chern number changing discontinuously
from $\pm2$ to zero beyond a certain critical hopping anisotropy remains a
common feature in the two cases.
