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Batteries increase ressistance when going bad/get old and get higher voltage drops during current draw. So while it might have normal voltage with normal draw(like turning pages), the worst case draw can be just enough to drop the voltage below the limit that the reader would consider it empty and turn off so it doesnt overdischarge the battery.I have measured an aura one, and worst case current draw is almost double current than page turn(250ma vs 450ma).
Maybe a sync happened during sleep and draw just enough current that the battery cant handle anymore, which made it turn off. Do you have wifi on when this is happening? If so , maybe try turning it off when not needed, might eleviate the problem a bit.
If its out of warranty i doubt kobo would even look in its general direction.
Without any more info or ideally testing, it looks like you need a battery change.
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