Electrical and thermal conductances of a quantum Hall bar reflect the topological structure of the incompressible bulk phase. Here we show that noise of the electrical current carried through the edge is evidence of the interplay between these two topological observables. Transport through a structured edge is modeled by a voltage-biased line junction made up of two counterpropagating modes associated with respective filling factors. Specifically, we focus on the edge of a ν =2 /3 fractional quantum Hall state. Noise is generated at a point distinctly separated from the hot spot (where most of the Ohmic dissipation takes place) and reflects the competition between ballistically carried downstream current and diffusively carried heat (which can propagate also upstream). We propose specific setups where our predictions can be measured.