  We theoretically explore the behavior of thermal transport in the topological
SQUIPT, in the linear and nonlinear regime. The device consists of a
topological Josephson junction based on a two-dimensional topological insulator
in contact with two superconducting leads, and a probe tunnel coupled to the
topological edge states of the junction. We compare the performance of a normal
metal and a graphene probe, showing that the topological SQUIPT behaves as a
passive thermal rectifier and that it can reach a rectification coefficient of
up to $~ 145\%$ with the normal metal probe. Moreover, the interplay between
the superconducting leads and the helical edge states leads to a unique
behaviour due to a Doppler shift like effect, that allows one to influence
quasi-particle transport through the edge channels via the magnetic flux that
penetrates the junction. Exploiting this effect, we can greatly enhance the
rectification coefficient for temperatures below the critical temperature
$T_\text{C}$ in an active rectification scheme.
