You shouldn't - manual says hose going into the heater core should be AT LEAST 29C/84F when the engine has warmed up. Easy to check if you have a thermometer. Much lower than that, then you likely have a bad thermostat or blockage as others have already suggested.
Based on what you said about the temp gauge sitting in the proper range, I'd guess it is more likely a problem with the temperature door. Our Aveo is not around for me to check, but book says you should be able to hear the door hit the stops by turning off the ignition, moving the temp control to cold, then moving it rapidly to hot. Dealer needs to check this out if you can't hear it.
The variability of the temperature is weird. Had these symptoms on a different car that turned out to be caused by air accumulating in the system from a small leak in a connecting pipe near the top of the engine. So, wouldn't hurt to confirm your dealer has actually pressure tested your system if they're unable to find the cause.