Hoorah, another CAI thread. I swear those 3 letters and some foam are the seemingly holy grail of unlimited, almost free power to some hahah.
Heating air up, isn't the same as heat soak. Heat soak specifically describes when latent energy moves from one medium to another. In this case, its the engine bay heat, moving into the air box / intake piping, and then into the intake charge
This is exasperated when the car is stand still for obvious reasons. A CAI works simply due to the partition slowing the heat energy movement from engine bay into the air immediately around the pod.
As mentioned, a CAI's main function i think, is not the temp drop at the compressor inlet from 50/80o to ambient (if setup is perfect with insulated intake to the turbo) but its ability to prevent heat soak when standstill; that is to provide more even temps and give a more consistent feel to the car. Every turbo car iv been in, takes a massive response hit after sitting still in a 40o day for a few minutes. Those with properly setup CAI's always seem to preform better.