  We uncover a useful connection between the integrated current noise
$S(\omega)$ and the topological band gap in dispersionless quantum states,
$\int d \omega [ \mathcal S^{\text{flat}}_{xx} + \mathcal S^{\text{flat}}_{yy}
] = C e^2 \Delta^2$ (in units $\hbar$$=$$1$), where $C$ is the Chern number,
$e$ is electric charge, and $\Delta$ is the topological band gap. This
relationship may serve as a working principle for a new experimental probe of
topological band gaps in flat band materials. Possible applications include
moir\'e systems, such as twisted bilayer graphene and twisted transition metal
dichalcogenides, where a band gap measurement in meV regime presents an
experimental challenge.
