The “background temperature” of the universe is already only about 2.8 K. Is that not cold enough for you?
Of course not. As cold as 2.8 K is, it’s a heck of a lot warmer than the 0.8 K achieved by the experiment. Fun stuff!
Problem is, as global entropy increases, things get warmer, not colder. The background temperature is now as cold as it is because a lot of the Universe’s energy is concentrated in small areas (we call them galaxies :^). As entropy increases, this energy (mass-energy, really) gets dispersed more evenly throughout space. Thus the “temperature floor” of the Universe will be higher than it is now (probably by less than a tenth of a degree K, as there’s a whole lot of nothing Out There.)
Also, in order for interesting quantum effects to occur in matter, there has to be matter- that is, aggregations of particles in a (relatively) small space. With particles dispersed more-or-less evenly throughout space (not perfectly evenly, but with a stochastic distribution) there’s no “matter” as we know it, but just a thin, thin fog of particles.
However, quantum fluctuations also occur in vacuum, so fun stuff can still happen.
Also, I’d like to remind everyone that entropy always increases in closed systems. The Universe may or may not be closed in this way. We just don’t know yet. For example, if the Universe is an infinite heatsink, then it can just disperse heat forever and never warm up. Local energy gradients would never break down (unless and until all energy sources in the ‘verse run down.) There are many ways in which the Universe could prove to be non-closed wrt energy.