Do zero-frequency modes contribute to the entropy?

Unruh's mixed state of a field with a zero-frequency mode, which purportedly contributes an arbitrarily large entropy for a given energy, is actually a mixture of pure states belonging to different systems, since the mean value of the scalar field is different for each pure state in the mixture. Considerations related to superfluidity support the idea that vanishing entropy is to be ascribed to a zero-frequency mode.