This is a potentially dangerous situation here. Diesel exhaust poisoning (carbon monoxide poisoning) initially are headaches, fatigue, dizziness, and flu-like symptoms. Chronic exposure can cause memory loss, confusion, hallucinations, and death.
If you are correlating your symptoms with the APU being on while you are closed up, it very well could be an exposure to carbon monoxide.
You should INSIST that service/testing be done on the APU to detect for problems. Put your request to them in writing just in case something comes up.
In the meantime, you will be forced to keep your windows open in order to prevent some serious consequences.
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