The Standard Model interacts with cosmology (as we know it) in at least two ways, although it is true that quantum mechanics simply can't be applied (not without some considerable, er, creativity) at the level of general relativity.
Firstly, our best large-scale model of the universe assigns a large proportion of the mass-energy of the universe to dark matter and dark energy. Those are particles or fields that the Standard Model currently doesn't contain, and any higher-energy extensions of the Standard Model get a big boost whenever they can churn out predictions for what dark matter or dark energy might be. Since those predictions have to match cosmological data, the Standard Model does intersect with cosmology there.
Secondly (and this is more fun), the Big Bang theory models the universe as having been infinitely dense in its past. The smaller the universe, the higher the density of mass-energy in it (given conservation of mass-energy), so that at some point the entire universe was small and high-energy enough for quantum mechanics to directly govern its entirety.
As an analogy, most cheap balloons have a little nub on top of them; when you blow them up, this results in a patch of thicker rubber at the top of the balloon. Now someone who has only ever studied inflated balloons should be able to look at the patch on top of the balloon and infer that, when the balloon was uninflated, it must have had an extra nub of rubber up there. In the same way, a really robust theory of the Big Bang and cosmic inflation should be able to wind the universe back far enough and generate high-energy predictions that we can then compare to our own high-energy experiments on Earth.
The recent BICEP2 cosmic microwave polarization results belonged in this category - although latest results suggest that the polarization in fact came from galactic dust rather than cosmic inflation, marking a rather sad day for what would have been an incredible result. But my favourite result of this sort is the baryon acoustic oscillation measurement of the flatness of the universe.
Basically, if you go back far enough, matter is so hot and dense that (1) light can't travel any distance without being absorbed and re-emitted, (2) the absorption and re-emission of light allows matter to jostle against itself to produce compression waves, just like sound. During that time, there would have been great oscillations moving throughout the universe creating some patches where matter is more dense and other patches where matter is less dense. Now imagine the very moment that the universe expands enough that this is no longer the case. Light is now free to travel throughout the universe, meaning that matter can begin collapsing into stars which are free to radiate. On the other hand, the great oscillations mean that at that very moment there must have been regions that are more dense and regions that are less dense. Gravity causes matter to be attracted to other matter, meaning that matter will keep disappearing from the less dense regions into the more dense regions, giving rise to a universe today where the vacuum of space is many million times more empty than a typical galaxy.
Now if galaxies (today) mark where matter was once more dense, and matter was once more dense where the oscillations made them clump up, then you can chart where galaxies are today and see if there is any kind of ordering to their position - and indeed there is! Galaxies are not uniformly scattered in space like chocolate chips in a cookie with the exact same distance between any of them, instead showing clumps and gaps (which is not to say they aren't randomly placed).
So on the one hand astronomers can go out looking for and measuring these clumps and gaps. On the other hand, the quantum mechanics boffins can make up a ton of equations telling us exactly when the universe would have expanded past that critical point, and exactly how large the oscillations must have been, and therefore how big those gaps must be today (after expansion).
And guess what? The gap measurements by the astronomers match up precisely to the predictions by the boffins! And it gets even better because this turns out to be a measurement of the curvature of space. Basically, if space is curved positively, far away things should look smaller than they actually are, and if it is curved negatively they should look larger. But the gaps look exactly as large as they should, which is exactly what scientists mean when they say that the universe is flat - and remember, it's quantum mechanics at very high energies and very small scales that predicts how large these gaps should be.
You can't make this stuff up. Either the universe is playing a cosmic prank on us, or quantum mechanics and cosmology confirm each other and confirm that this is scientifically the best way of explaining the universe.
