Probably what is meant is that in an ideal world 3 mins. is required before shutdown, but for operational reasons eg. small airfields with minimal taxi time from runway to ramp then 1 min. is acceptable.
The cooldown time is there to allow the engine spools and casings temperatures to equalise.
FWIW the GE90 requires 10 mins. which is difficult to achieve at some fields.
The most noticeable effect of shutting down from a high power setting without allowing the core to cool is seized rotors (the good old P and W JT9D was good for that) to high vibration on start up (Typical of GE CF6 and GE90 (bowed rotor))