I've worked with multiple foamcore thickness, to prevent warping I would suggest 1/2 or thicker. you won't regret it! I have used the standard foamcore and experienced loads of warping on a garage I built, however once I utilized the 1/2 thickness no worries!
I'm also building a house out of gator board, although it is much sturdier at a smaller thickness, it requires sanding prior to painting and is quite expensive where I am and almost impossible to procure. hth! and good luck in whatever material you choose to use! I look forward to seeing some construction photos!