While usually we write Nernst equation using concentrations, in fact it should contain not concentrations, but activities of ions. In fact everywhere you use concentrations (like all equilibrium calculations) you should use activities, not concentrations. For very diluted solutions, with ionic strength close to 0, activity equals concentration, but the higher the ionic strength, the higher the difference (to make things more difficult, the difference between activity and concentration doesn't change in an easy to predict and calculate way).
Sure, there exist a standard solution with copper(II) activity equal 1, and then e0=1 would hold. However, even then it doesn't mean potential is temperature independent, as changing temperature of the solution means changing the volume, changing the concentration, changing the activity - and it is will be no longer 1.