If ambient temperature, at (initial) startup, is less than spec, whether a startup failure might be logged, I expect, would much depend how far into the startup the device manages to perform. I.e. a very early failure might not get far enough to log anything, or even display anything to the console.
Possibly, the device might somehow get far enough into its startup process, to internally note its aborting startup which might be logged "normally" in a later successful startup. (For the last, I have in mind a special diagnostics card I had, decades ago, that you could plug into a PC card slot which might display POST codes, and if PC didn't boot, that code might indicate what hardware was failing during initial power on. Keep in mind, if something fails during power on, depending on what's the problem, can very much limit the device from, somehow, indicating the nature of the issue.)
The forgoing should not be confused with a quick power cycle startup, where the device may still be warm from prior operation.
