The starter and start duty cycle limitations on APU's vary with the APU, and the installation (with the aircraft), and in some cases, with the operator. The same limitations do not apply across the board to all APU's.
On the first start attempt, the start motor heats up, It's given a minimal cooling time, and a second attempt allowed. When this second attempt occurs, the start motor will not have cooled to its original temperature (where it was when the fist start attempt occurred). The heat increase with the second start attempt, then, is cumulative. In some aircraft, a third start attempt may be allowed, while in others it may not; if another start attempt occurs, the starter temperature increases even further.
If the APU attempts to light off and then fails, it will have an internal temperature increase which is not uniform, or "normalized" throughout the gas path in the APU. Successive start attempts will worsen this condition. Time is required for the temperature to decrease.
Remember that the APU, just like the main engines, is composed of many components which expand and contract at different rates. The APU gets less airflow once the start has failed, than a wing mounted engine. It has less ventilation, and less cooling.
Additionally, if fuel was admitted to the APU during the start process (depending on where the start attempt cut off), fuel may be pooled in burner cans, and could lead to a fire with successive starts; part of the time between starts is a draining period.
Another potential problem, depending on the power source used during the APU start, is ship battery depletion. Successive starts can drain the aircraft batterie(s), and can lead to a large current draw from other sources, through the ship battery. This can cause a battery overheat or even a potential fire, or may damage the battery.