Just my two cents.
If you see the details of air intake systems in new cars, you can see that the air going to the intake manifold is actually coming from outside the engine bay. At least in new corolla this is the case. So you can say that cold air is flowing to intake manifold in the stock system.
According to me, the only difference in so called CAIs is that the volume of air is increased. More air can flow through them.
I have one more doubt. Generally after market intake system have metal pipes (hoses) and stock intake systems have rubber pipes. I think metal pipes get more heated inside the engine bay than rubber pipe. If this is the case then isn't the air flowing through these metal hoses warmer? May be experts can answer this.
thanks,