The dog's helmet was open in the movie so it must have been able to breathe. It was kept by the Collector in a cell. It's not like the dog is going into space any time soon: so, why does it still have a space suit on? Even at the end of the movie post credit, Cosmo was running around outside of its container without issues so we know the cell did not contain any kind of special air that the dog needs.
On a side note, I see an issue with potty with that space suit.
In the Guardians of the Galaxy movie, Cosmo is an exhibit of the Collector. In the comics, Cosmo was the "sheriff" of Nowhere, so there is a bit of a medium conflict. In the comics, he was still wearing his Soviet-era suit even though he had people to help him, support him and provide him with whatever technology he might have needed.
Cosmo is a dog from a Soviet Space Program who was whisked away into an experimental wormhole. When he arrived on the other side he found his intelligence greatly increased and he was now blessed with incredible psychic abilities.
However, living on Nowhere (before he was captured by the Collector) was now very far from home and nostalgic. While he could get other clothing or wear none at all, he is homesick for his home planet and wears the only artifact he can assure has come from Earth.
As an exhibit of the Collector, odds are, knowing the Collector's habits, the suit has been modified to meet any needs his "exhibits" might have. If he didn't want to change the suit in any way, he may have had robots or servants to "clean up" after Cosmo. Since the suit came with the dog, from the Collector's perspective, it adds to Cosmo's "value" since the suit was from an early civilization's space program.
In the movies, he seems bent on staying with the Collector. In the comics, he has run off to work with the Guardians of the Galaxy.
