  The $\alpha$-$T_3$ lattice, an interpolation model between the honeycomb
lattice of graphene($\alpha=0$) and the dice lattice($\alpha=1$), undergoes a
topological phase transition across $\alpha=1/\sqrt{2}$ when exposed to a
circularly polarized off-resonant light. We study Berry phase mediated bulk
magnetic and anomalous thermoelectric responses in order to capture the
topological signatures of a driven $\alpha$-$T_3$ lattice. It is revealed that
both the Berry curvature and the orbital magnetic moment associated with the
flat band change their respective signs across $\alpha=1/\sqrt{2}$. The
off-resonant light distorts the flat band near the Dirac points when
$0<\alpha<1$ which eventually introduces two distinct well separated forbidden
gaps of equal width in the quasienergy spectrum. The orbital magnetization
varies linearly with the chemical potential, in the forbidden gaps. The slopes
of the linear regions in the orbital magnetization are closely related to the
respective Chern numbers on either side of $\alpha=1/\sqrt{2}$. We find that
the slope for $\alpha>1/\sqrt{2}$ is approximately two times of that for
$\alpha<1/\sqrt{2}$ which essentially indicates a topological phase transition
across $\alpha=1/\sqrt{2}$. However, the anomalous Nernst coefficient vanishes
when the chemical potential is tuned in the forbidden gaps. The anomalous Hall
conductivity in the forbidden gap(s) approaches different quantized values on
either side of $\alpha=1/\sqrt{2}$. All these topological signatures can be
observed experimentally.
