Coherent caloritronics, the thermal counterpart of coherent electronics, has drawn growing attention since the discovery of heat interference in 2012. Thermal interferometers, diodes, transistors and nano-valves have been theoretically proposed and experimentally demonstrated by exploiting the quantum phase difference between two superconductors coupled through a Josephson junction. So far, the quantum-phase modulator has been realized in the form of a superconducting quantum interference device (SQUID) or a superconducting quantum interference proximity transistor (SQUIPT). Thence, an external magnetic field is necessary in order to manipulate the heat transport. Here, we theoretically propose the first on-chip fully thermal caloritronic device: the phase-tunable temperature amplifier (PTA). Taking advantage of a recently discovered thermoelectric effect in spin-split superconductors coupled to a spin-polarized system, we generate the magnetic flux controlling the transport through a temperature-biased SQUIPT by applying a temperature gradient. We simulate the behavior of the device and define a number of figures of merit in full analogy with voltage amplifiers. Notably, our architecture ensures almost infinite input thermal impedance, maximum gain of about 11 and efficiency reaching the 95%. This concept paves the way for applications in radiation sensing, thermal logics and quantum information
