The old HPAK 34401As boot up quickly, but take some time (2~3hrs) to stabilize since they have no fan. All the newer bench types we have (KS34465A, DMM6500, DAQ6510, SDM3065X) take some time to boot up and usually stable within 0.5~1hr and have a fan.
Would be a nice if the new bench DMMs had a feature to allow critical internals (like LM399) to remain powered and the display and other functions in limbo or unpowered, but this might not be useful as without the fan on and all DMM circuits powered the internal temperature distribution will change.
We measured a temperature of 2~3 C above ambient on some DMMs just at the banana sockets by means of plugging in a dual banana plug supported precision resistor which we had attached a thermistor to measure the resistor temperature. This alone indicates the actual DMM internal banana sockets are higher in temperature and the internals even higher.
We're sure this has crossed the minds of the KS and Keithley DMM engineering staff before, as it's an obvious benefit, however this might be the reasoning as to why such hasn't been implemented at least in the higher end precision DMMs.
Best