If the Universe is infinite, would heat death be impossible? If the Universe is infinite, one would imagine that if we kept moving through space, we'd constantly find newer and newer galaxies and objects forever. This would imply that there is an infinite amount of matter in the Universe.

The heat death theory states that, eventually, there will be no free energy left in the Universe. Stars would no longer form, black holes would eventually evaporate, and the Universe would be effectively dead. But if there's an infinite amount of matter, won't there always be more gas clouds that haven't formed stars, more stars that have yet to become black holes, etc.? If there is an infinite amount matter and energy, shouldn't more activity be possible no matter how many stars go out or how many black holes evaporate?

One might say that with enough time, all matter in the Universe would still run dry. But if there is an infinite amount, there should always be areas in which activity is still going on...right?

Is what I'm saying correct? Does the heat death of the Universe contradict the theory that it is infinite?
Heat death of the Universe doesn't mean that there is no energy in it, it means it has reached thermodynamic equilibrium and hence no useful energy in it. This could in theory occur in an infinite Universe.

Of course thermodynamics is statistical in nature, so an argument could be made that some extremely unlikely event in a region of the Universe could destroy thermal equilibrium in that region and that the larger the Universe the more likely that some region like that exists.