As it's said here(APS-3200 training manual (pg 268)):
we need 1 hour for cooling after 3 consecutive attempts for starting the APU. But why?
why not after each attempt? and why 1 hour?
is it valid for all APUs?
What about the jet engines of aircrafts? is there something like that for them?
That is typical of any electrically started component. During qualification testing of the APU, tests will be done to measure the heat build-up in the starter motor to determine how many start attempts it takes to get the motor's temperature to a critical level. Other tests determine how long it takes to cool down enough to operate again, after it gets to that temperature.
In a nutshell, the limits are to protect the APU's starter/generator from heat damage if multiple start attempts are needed. Turboprop engines with battery powered starter/generators will have similar limitations.
Jet engines typically use bleed air driven turbine drive starters to start, and may have similar, if somewhat looser, limitations compared to an electrically started engine (to control the starter's gearbox oil temperature).
For example, on the CF34, you're allowed to crank for 90 seconds, and then there's a 10 second cool down after the 1st two attempts, and a 5 minute cool down after each subsequent attempt, up to 5 tries (after that, it's expected that you've given up and called for maintenance). Other than that, no limitations.
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.
