Maybe I didn't explain it well enough before.
Cold air requires less work energy to compress a given mass than hotter air.
Because it is colder, you can put more heat energy into it without melting the HP turbine.
Because of the lower compressor requirement, less turbine work and expansion is required to power the compressor.
Hence proportionally more gas energy is available to turn the fan or expand through the primary nozzle (= greater thrust per weight of fuel = efficiency)