Topological effects in thermodynamics*

We consider entropy and persistent currents induced by topological phases in multiply-connected open quantum systems. We prove a strong form of the Nernst theorem (third law of thermodynamics) for "fully open" quantum systems: the entropy goes strictly to zero as temperature approaches absolute zero. The conventional formula for the heat current is shown to be problematic for persistent currents, implying a divergent entropy current as temperature goes to zero, in contradiction to the third law. The apparent paradox is resolved through the inclusion of a topological work term in the first law corresponding to the "persistent work" done in establishing the topological fields.