This is not a fault, hairdryers do not actively cool air. The cool shot function differs between DC & AC motor hairdryers.

On AC motor hairdryer - It does not actively cool the air, but the heaters are able to completely shut off, so only room temperature air is being pushed through. Much like a cool breeze or the wind, when the air is blown, it feels ‘cooler’ even though the temperature is ambient.

DC motor - A small section of the heater must remain on, so it is slightly warmer than ambient/room-temp.