Should an air conditioner run all day and does it harm the equipment?

An AC should not tyipcally run all day. However, with extreme heat and the thermostat turned down low, the AC will run for extended periods. Typical rule of thumb is not to turn the thermostat more than 20 degrees less than the outside temp. When the temp is 100 degrees outside, the AC will have to run for extended periods to get the inside temp below 80. This varies much depending on the type of equipment you have, insulation in the home, tonage, SEER, etc. If the AC is running for extended periods, make sure that it is cooling properly. If the inside temp is not dropping, call a professional.