This seems orthogonal to me. Tcold for the efficiency is the exhaust temp, not the intake.
For pure efficiency, it is actually beneficial to capture waste heat in the exhaust to heat the intake air. But it destroys your power density, for the reasons already mentioned (hot air has less mass at the same pressure). I believe some automotive engines actually do this under light load conditions, but you wouldn't see it in F1.
Car engines are heat engines, they extract mechanical work from differences in temperature. Heat engine efficiency is n = 1 - sqrt(Tcold/Thot)
If Tcold, the cold end, isn't cold enough, then you lose efficiency, and thus horsepower. A badly overheating engine loses a lot of power. If you can cool the compressed air back down to ambient, then your engine produces more power.