  We address the problem of thermodynamic equilibrium with constant
acceleration along the velocity field lines in a quantum relativistic
statistical mechanics framework. We show that for a free scalar quantum field,
after vacuum subtraction, all mean values vanish when the local temperature T
is as low as the Unruh temperature T_U = A/2\pi where A is the magnitude of the
acceleration four-vector. We argue that the Unruh temperature is an absolute
lower bound for the temperature of any accelerated fluid at global
thermodynamic equilibrium. We discuss the conditions of this bound to be
applicable in a local thermodynamic equilibrium situation.
