Maximum entropy merely implies that the temperature everywhere is the same (any other situation would necessarily have a lower entropy). You could in theory have maximum entropy at 1000K. Our universe has a ton of empty space, and not all that much energy, so the temperature at which it equilibriates is very low. It's also expanding, so the hypothetical equilibrium temperature is decreasing all the time.
It's also worth noting that the entropy can be very large, even if the temperature is absolute zero. (You just need a system with a lot of different ground-states that all have the same energy.)
I've wondered about this. How can something which is having a cold death (zero kelvin) also have maximum entropy? But I am not at all an expert.