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Vacuum energy has zero-point energy. Does this imply that vacuum energy [has] an average temperature of 0° Kelvin?

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There are various ways to define temperature, but a simple way to understand it is that some medium has a temperature greater than zero if heat can flow from that medium to a colder medium.

But even though the vacuum does indeed have a zero point energy, that energy is still the lowest energy state possible. So there is nothing colder than the vacuum where the energy can flow, and that means the temperature of the vacuum is zero despite the zero point energy.

We get a similar, and simpler, phenomenon with liquid helium. When you cool helium down to absolute zero it still has a vibrational zero point energy, so even at absolute zero its energy isn't zero. In fact that zero point energy is great enough to keep helium a liquid even at absolute zero. Despite this the temperature is zero simply because the zero point vibrational energy in the helium can't flow anywhere colder.

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