  The stability of $\nu=1/3$ Fractional Chern Insulator (FCI) phase is analysed
on the example of checkerboard lattice undergoing a transition into Lieb
lattice. The transition is performed by the addition of a second sublattice,
whose coupling to the checkerboard sites is controlled by sublattice staggered
potential. We investigate the influence of these sites on the many body energy
gap between three lowest energy states and the fourth state. We consider cases
with different complex phases acquired in hopping and a model with a flattened
topologically nontrivial band. We find that an interaction with the additional
sites either open the single-particle gap or enlarge the existing one, which
translates into similar effect on the many-particle gap. Evidences of FCI phase
for a region in a parameter space with larger energy gap are shown by looking
at momenta of the three-fold degenerate ground state, spectral flow, and
quasihole excitation spectrum.
