If we had an infinite space constructed in a finite time, then surely there would be a history of the sky getting hotter as more radiating matter is included in

If we had an infinite space constructed in a finite time, then surely there would be a history of the sky getting hotter as more radiating matter is included in our causal horizon (light cone). Instead we see a sky at a near homogeneous temperature, meaning that every part of the Universe must once have been very close to every other part for a time sufficient to reach equilibrium, and then cooled down. The temperature history of the Universe appears to speak against infinite space.