I think "Hot" start has two entirely different meanings. The first hot start which we all dread is the "um gee im sorry boss but we need a new engine cause i just melted the other one" or a hot start just after shut down, when there is still quite a bit of residual heat still in the engine. The second hot start is not that much to worry about and is quite easy to do but as mentioned it does have a fair ammount of current drain on the batteries due to the extra motoring of the engine to bring it down to an acceptable turbine temp before inducing the fuel to avoid the 1st example of a dreaded hot start, hence why on short turn arounds, you or your company may decide to leave the right eng running so you get that extra kick to help it over the hump. Honestly on a pt6 powered aircraft, with todays batteries i have not yet had a hot start (1st), and even when doing a non assisted hot start (2nd) the difference between the peak itt of the two is usually 20 to 40 deg c. As for the twotter knew of companies that would transpose the engines from left to right at halve life ttis so that the engines did not cycle out due to one engine being constantly started and shut down all the time. As for it being the right engine that is left running, most turbine twins that i know of load from the left with exception of the co pilot. Hot start (2nd) on a garrett is a hole different ball game.