While we discuss the thermodynamics of a theoretical infinite plane, we should not forget that the real astounding failure this hypothesis has is that the Sun, stars and other celestial objects are just not in the right place. A hypothesis that really fails to predict that the stars appear to rotate around two points in space corresponding to the North and South poles is not worth too much elaboration. And we could and have talked for days about the whole list of errors that this hypothesis has with regards to the placement, brightness and apparent size of the celestial objects.
Anyway, you would have to have an infinite amount of mass just appearing from nowhere in a perfect plane to have this hypothesis work. Thermodynamics, gravitation, plate tectonics or anything else that disturbs the perfect initial plane will create an imperfection big enough to start the crumbling of the infinite plane into pieces that, with the help of gravity, will attract other pieces and in no time the perfect infinite plane will become, (you guessed it) a bunch of spherical planets and stars.
Remember, at planetary scales the solid objects do not have the consistency of steel, they rather have the consistency of wet clay.