  Superconductors are known to be excellent thermal insulators at low
temperature owing to the presence of the energy gap in their density of states
(DOS). In this context, the superconducting \textit{proximity effect} allows to
tune the local DOS of a metallic wire by controlling the phase bias ($\varphi$)
imposed across it. As a result, the wire thermal conductance can be tuned over
several orders of magnitude by phase manipulation. Despite strong implications
in nanoscale heat management, experimental proofs of phase-driven control of
thermal transport in superconducting proximitized nanostructures are still very
limited. Here, we report the experimental demonstration of efficient heat
current control by phase tuning the superconducting proximity effect. This is
achieved by exploiting the magnetic flux-driven manipulation of the DOS of a
quasi one-dimensional aluminum nanowire forming a weal-link embedded in a
superconducting ring. Our thermal superconducting quantum interference
transistor (T-SQUIPT) shows temperature modulations up to $\sim 16$ mK yielding
a temperature-to-flux transfer function as large as $\sim 60$ mK/$\Phi_0$. Yet,
phase-slip transitions occurring in the nanowire Josephson junction induce a
hysteretic dependence of its local DOS on the direction of the applied magnetic
field. Thus, we also prove the operation of the T-SQUIPT as a phase-tunable
\textit{thermal memory}, where the information is encoded in the temperature of
the metallic mesoscopic island. Besides their relevance in quantum physics, our
results are pivotal for the design of innovative coherent caloritronics devices
such as heat valves and temperature amplifiers suitable for thermal logic
architectures.
