I reckon it might. While I don't claim to have any knowledge of computer science (I am a total klutz with it) I have had the odd occasion where I was trying to trigger something in a game with a random chance, and found I was more successful if I did it just after switching the console on than if I'd been playing for a while.
I'm under the impression that the processor is what handles the generation of random numbers, and I'm also under the impression that its temperature can have a slight effect on these values. Is this true? Is this an actual thing that can happen?