  We consider a thermodynamic framework to quantify instrument incompatibility
through a resource theory subject to thermodynamic constraints. In this
resource theory, we use the minimal thermalisation time needed to erase
incompatibility's signature to measure how incompatible an instrument is. We
show that this measure has a clear operational meaning in some work extraction
tasks, thereby uncovering the thermodynamic advantages of incompatible
instruments. We further analyse the possibility and impossibility of extending
the time for incompatible signature to survive under general evolution.
Finally, we discuss the physical implications of our findings to measurement
incompatibility and steering distillation.
