Well, I have no experience of the big ones and have only operated PT6's in Twotters and Sheds plus the shaft turbine engines (Allisons?) in the J31.
Apart from sticking to the manufacturer's manual (obviously) the principle I have always been taught is the cooler the peak turbine temberature on start the better. Thus TBO's which in my case have always been determined by computer analasis of repeated recording of engine parameters in flight are maximised. - Well this was the theory. To minimise turbine start temperatures the power supply to the starter motor must be as good as possible. When starting these engines using on-board batteries, the batteries are operated virtually to their limits. Larger batteries than absolutely necesary will of course provide a weight penalty. The ideal way of starting is thus to use a GPU which can easily cope with the starting requirement and minimises the temperature spike on start up. If this is impractical or too expensive, then a GPU start should be used if possible for the first start of the day. This is because the battery voltage will have decayed slightly when parked overnight.
When a battery start is needed, the first engine is started, and once it is up to an appropriate speed the starter-generator is switched to generate to re-charge the batteries. The charging current is monitored and when this has fallen below a specified figure the battery is deemed fit for another start cycle. The charging generator (only on the Twotter if I remember correctly) is then switched off again and the second engine is started.