Why we need 1 hour for cooling after 3 consecutive attempts for starting the APU?
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.