Everyone will have their own preferences regarding leaving on or shutting off, but I
would turn my T3, (and the T2 and T1 before it), on first thin in the morning and
off just before going to bed - 16 hours per day average.
To be sure, it was not being played all that time and the hard drive would go to sleep
after a period of time - the keyboards do not consume a lot of power (60 watts for
Tyros and half that for the PSR's).
Jørgen has mentioned the most likely solution and the problem of shutting down while playing.
My MOX has the auto shut down feature and I have disabled it because you stand the chance
of loosing music work in process if you have not saved it and you leave the keyboard for a bit.