Is the warm air blowing into the cooler or out of the cooler? If it's exhausting warm air from the fan to the outside of the cooler, that's perfectly normal and expected.
Most of these sorts of electric coolers use a Pelletier junction as a heat pump. If the polarity of the electric power to the junction is switched, the junction operates backwards and moves heat the other way (which, for the cooler, makes the inside warm and the outside cool). Many have a switch or a reversible cord to enable this so you can use the cooler as a warmer to keep your casserole or whatever nice and toasty en route to the potluck. If the inside of the cooler is getting warm, your problem is probably that the cord is reversed or this switch is in the wrong position for what you want.
If the lighter socket it's plugged into were somehow wired backwards, that would have the same effect. That's a pretty unlikely thing to come up.
