You will not hurt the amplifier by letting it idle to reduce the internal cabinet temperatures but the main internal heat source is no longer operational. Glass tube amps have a small heat source and they will not have much temperature mass. The do relay on the fans to route the heat out of the chassis.
Metal tetrodes run at a much lower temperature, typically below 50 degrees C, and generally have the vent holes right above the tube. Natural heat convection will vent the anode with out the fan running.
My ACOM has a single Svetlana ceramic metal tetrode with the vent holes above it in addition to a built in fan.
It appears to be hotter than 50C when I am long winded even at a modest 400W.
I'll have to check with the temp gun next time.
I'll continue to let it idle a while before shut down since it doesn't hurt anything other than a penny in electric.
My temp runs 60 degrees C after a long QSO via temp gun reading at the chassis vent.
Buddy's 4x10 grid driven amp runs at 150 F (65 C) exhaust air temp, while blathering away on ssb, using ptt / footswitch. Once released, it takes a full minute to get back down to 90 F (32 C) ...where it remains. The 90 F is coming from the cathode sucking a lot of fil power. I just found it easier on the hb amps, to switch to standby, then kill the B+, then the fil xfmr, then everything is off. Then let blower run for a few minutes, before shutting off the blower.
On RTTY/ FM / DATA modes / AM, average plate current is a lot higher..and so is exhaust temps. Tank coils / band switches will also dissipate more heat, since quadruple the watts is being dumped into them. It's just I squared X R..or in this case, RF current squared X the ESR (rf resistance) of the tank coils. RF current in the tank coil is aprx plate current X input loaded Q of the PI net..+ 10%. I was surprised at just how hot tubing coils got on 15m, with a brief cxr applied, then removed, then touched the 15m tank coil, yikes, fn hot. I would have thought the heat would have migrated into the 20m portion of the same 20-10m tubing coil, but it doesn't. 20m end of same coil runs at ambient room temp. It's all localized heat. Then toss in still more heat from the plate xfmr etc.