  Electrons on a two-dimensional (2$d$) lattice which is exposed to a strong
uniform magnetic field show intriguing physical phenomena. The spectrum of such
systems exhibits a complex (multi-)band structure known as Hofstadter's
butterfly. For fillings at which the system is a band insulator one observes a
quantized integer-valued Hall conductivity $\sigma_{xy}$ corresponding to a
topological invariant, the first Chern number $\mathcal{C}_1$. This is robust
against many-body interactions as long as no changes in the gap structure
occur. Strictly speaking, this stability holds only at zero temperatures $T$
while for $T>0$ correlation effects have to be taken into account. In this
work, we address this question by presenting a dynamical mean field theory
(DMFT) study of the Hubbard model in a uniform magnetic field. The inclusion of
local correlations at finite temperature leads to (i) a shrinking of the
integer plateaus of $\sigma_{xy}$ as a function of the chemical potential and
(ii) eventually to a deviation from these integer values. We demonstrate that
these effects can be related to a correlation-driven narrowing and filling of
the band gap, respectively.
