Mark,
which particular tanks are you looking at - old, or modern? All larger engines need to crank up to speed before they fire up - diesel engines especially. If you're talking about gas turbine tanks, then they really have to spin up to speed before power kicks in.
Keeping the engine warm would reduce the start up time ( 5-10 mins running an hour would do this ). Most tanks now have an APU, which keeps the batteries charged & radios working, without running the thirsty main engine - the cooling water from this keeps the main engine hot & ready to go.
Climate has a strong influence on start up times - if it's really cold, then cranking times can be extended, and you have to use starting aids - not good for the life of an engine!
I was involved with designing & testing auxy. gens sets for the military, and some had to be started at -15C!
Dave