Unbreaking X doesn't stop time-based LP cost. Doesn't seem to affect per-hit LP cost either but it's hard to say for sure since the time intervals for taking LP seem to be a bit random. Either that or I'm getting an inconsistent tick rate on my tiny test world, but I can sit there mashing right click as fast as I can for the divination sigil and sometimes see my LP jump down by 100 instead of the usual 50 at a time.
I'm guessing it does nothing, since I just lost 150 LP in the time it took me to spawn and kill a mob in 1 hit and check again.