It is shown that the Unruh effect, i.e. the increase in temperature indicated by a uniformly accelerated thermometer in an inertial vacuum state of a quantum field, cannot be interpreted as the result of an exchange of heat with a surrounding gas of particles. Since the vacuum is spatially homogeneous in any accelerated system its temperature must be zero everywhere as a consequence of Tolman's law. In fact, the increase of temperature of accelerated thermometers is due to systematic quantum effects induced by the local coupling between the thermometer and the vacuum. This coupling inevitably creates particles from the vacuum which transfer energy to the thermometer, gained by the acceleration, and thereby affect its readings. The temperature of the vacuum, however, remains to be zero for arbitrary accelerations.