Directed Energy Weapon Eater
one could armor something we wish to protect with the bottomless heatsink to protect it against directed energy weapons. So long as we can absorb all of the incident energy and the heat transfer rate to the bottomless heatsink is high enough, we can prevent the armor from melting. For more fun, we can use said absorbed heat to generate power. This probably won't protect your crew from getting irradiated by particle beam weapons without impractically thick armor
Many industrial processes require heat to be removed. In chemical plants we wish to remove heat so that things may be condensed. A bottomless heatsink that can keep the external temperature of a spacecraft at 3K could make attaining cryogenic temperatures easy. We also get nifty things like self powered air separation plants. Because we have cheap cryogenics we could produce amorphous metal like we produce steel.
Keeping superconductors cool
With a bottomless heatsink, you don't need room temperature superconductors. Whether this makes superconductors more common or just a novelty for levitating cup-holders will depend on how cheap the bottomless heatsink is and how fast it can transfer heat. Superconducting power lines and maglevs might not necessarily happen just because of the issues with dealing with extremely cold things. For example, we can't just spray our bottomless heatsink on wires to make them superconduct and hang them up on power poles because they'll accumulate a layer of ice. Although if you can keep a spaceship's hull at 3K then you can probably also keep a bunch of loops of superconductor cool in space to trap antimatter orbiting in planetary magnetospheres
keeping quantum things cool
In addition to superconductivity we can exploit low temperatures for other weird quantum phenomenon like superfluidity, quantum computers, bose-einstein condensates. Determining the applications of such phenomenon is its own question, although I will point out that bose einstein condensates have been used to slow light down to 25 km/h.
we can store medicines which need to be refrigerated passively. Blood, tissue, organs, could also be stored this way too. This removes the difficulty with keeping cryonics patients cool.
The ultimate beer koozie
It is said that if you give man the hand of god, he will almost immediately use it to scratch his behind. If the material is cheap enough it could be used to keep drinks cold passively. We can of course scale this up to a beer cooler which can keep drinks cold almost forever. The point here is that if your material is cheap enough, there are plenty of mundane applications. There are likely ample applications in cooking, with ice cream machine being the first thing that comes to mind. Because we can passively achieve cryogenic temperatures, we can make an LN2 ice cream machine that fits in a typical home kitchen.