We theoretically put forward the concept of a phase-controlled superconducting heat-flux quantum modulator. Its operation relies on phase-dependent heat current predicted to occur in temperature-biased Josephson tunnel junctions. The device behavior is investigated as a function of temperature bias across the junctions, bath temperature, and junctions asymmetry as well. In a realistic Al-based setup the structure could provide temperature modulation amplitudes up to ~50 mK with flux-to-temperature transfer coefficients exceeding ~125 mK/Phi_0 below 1 K, and temperature modulation frequency of the order of a few MHz. The proposed structure appears as a promising building-block for the implementation of novel-concept caloritronic devices operating at cryogenic temperatures.