  The layered topological insulator MnBi$_2$Te$_4$ has attracted great interest
recently due to its intrinsic antiferromagnetic order, potentially hosting
various topological phases. By temperature-dependent infrared spectroscopy over
a broad frequency range, we studied the changes in the optical conductivity of
MnBi$_2$Te$_4$ at the magnetic ordering temperature. The temperature dependence
of several optical parameters reveals an anomaly at the magnetic phase
transition, which suggests the correlation between the bulk electronic band
structure and the magnetism. We relate our findings to recent reports on the
temperature dependence of the electronic band structure of MnBi$_2$Te$_4$.
