There isn't much here other than the AFM procedure. I'll give this more thought when I get a chance.
Everything here is from the references shown below, with a few comments in an alternate color.
A hot start even N1, N2, or TGT above limits.
(1) POWER LEVER (Affected Engine) . . . Operate at reduced setting
(2) Airplane . . . Land as soon as practicable
(1) POWER LEVER (Affected Engine) . . . IDLE
(2) FUEL CONTROL (Affected Engine) . . . OFF
(3) Refer to 04-08-10, Engine Shutdown in Flight.
Gulfstream GVII-G500 Airplane Flight Manual, Revision 4, August 29, 2019
Gulfstream GVII-G500 Production Aircraft Systems, Revision 3, July 15, 2019
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