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As defined by Wikipedia:

The heat death of the universe is a suggested ultimate fate of the universe in which the universe has diminished to a state of no thermodynamic free energy and therefore can no longer sustain processes that consume energy (including computation and life). Heat death does not imply any particular absolute temperature; it only requires that temperature differences or other processes may no longer be exploited to perform work.

Does it even make sense to describe temperature in the system described? If so, would it be a very cold system or a very hot system?

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Yes, and it would be very cold. The paper "finite temperature in a deSitter universe" explains that the cosmological constant (if it is really a constant) creates a "horizon" that acts somewhat like an inside-out event horizon: objects that get too far away from you are unreachable. This Horizon will radiate Hawking radiation at an extremely low temperature of 10^(-30)K; the wavelength range of light this corresponds too is the same order of magnitude as the horizon's radius (also the case for black holes). It's conceivable that some relatively compact system could have an excited quantum state so close to it's ground state that it is thermally accessible even at this extremely cold temperature. Thus, there is still some form of heat swimming around. However, there is no reservoir colder than this temperature to dump heat into so this heat can't be converted into useful energy.

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  • $\begingroup$ Well, am I surprised to learn this. I guess I was misled by the 'heat' in 'heat death' and the fact that it's being continuously belched into the universe. $\endgroup$ – Vandermonde Jul 5 '15 at 19:06
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Does it even make sense to describe temperature in the system described?

Not in thermodynamic sense. In thermodynamic sense, temperature is a property of state of thermodynamic equilibrium. To have a system in such a state, one needs to make sure neither heat transfer nor any macroscopic changes happen to the system.

How would you do that for universe? We don't know what happens at the borders, and there may not be any. The current observations do not suggest bodies in the universe tend to equilibrate. For example, stars are very hot, the matter in between much colder. This inhomogeneity may go on forever.

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    $\begingroup$ It may be more homogeneous than you think. The vast majority of the photons flying through the universe are those of the CMB. They're pretty cold, and will get colder as space continues to expand. And of course the era of star formation can't last forever. $\endgroup$ – PM 2Ring Sep 12 '18 at 8:33

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