  We study the Hall conductivity in single layer gapped Dirac fermion materials
including coupling to a phonon field, which not only modifies the
quasi-particle dynamics through the usual self-energy term but also
renormalizes directly the gap. Consequently the Berry curvature is modified. As
the temperature is increased the sign of the renormalized gap can change and
the material can cross over from a band insulator to a topological insulator at
higher temperature (T). The effective Chern numbers defined for valley and spin
Hall conductivity show a rich phase diagram with increasing temperature. While
the spin and valley DC Hall conductivity is no longer quantized at elevated
temperature a change in sign with increasing T is a clear indication of a
topological crossover. The chirality of the circularly polarized light which is
dominantly absorbed by a particular valley can change with temperature as a
result of a topological crossover.
