I want to bring a bottle of Champagne on the plane in my checked luggage. Will the pressure affect it?
Unless you're flying Third World Air, there should be no issue.
As far as I know, the cargo compartments in most commercial airliners are pressurized. On jets that carry pets, they also are heated.
Although the plane's interior goes through small changes in atmospheric pressure (which cause your ears to pop and sometimes makes flimsy shampoo bottles leak through the cap), your rigid Champagne bottle – notwithstanding the high internal pressure of the carbon dioxide – will be fine.
Theory aside, take it from somebody who's flown with sparkling wine more times than he cares to count. I've never lost a drop – or a bubble.