Does a universe experiencing "heat death" have a temperature? 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?
 A: 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.
A: 
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.
