Heat transport in a qubit–oscillator junction described by the quantum Rabi model is investigated. Upon variation of temperature, bias on the qubit and the qubit–oscillator coupling strength, a rich variety of effects is identified. For weak coupling to bosonic heat baths, transport is essentially controlled by the qubit–oscillator coupling g which defines a Kondo-like temperature
. At temperatures much lower than , coherent heat transfer via virtual processes yields a T3 behavior in the linear conductance as a function of T, modulated by a prefactor determined by the junction parameters and unravelling its multilevel nature. In particular, a coherent suppression of the conductance arises in the presence of quasi-degeneracies in the spectrum. For , sequential processes dominate heat transfer and a scaling regime is found when quantities are scaled with . The conductance as a function of the bias on the qubit undergoes a transition from a resonant behavior at weak qubit–resonator coupling to a broadened, zero-bias peak regime at ultrastrong coupling.