I honestly have never once checked the temp of my lye solution. I mix my lye & let it sit in the sink while I organise my oils & anything else I'm using. Once I have every thing together, I add the lye to the oils and off I go. The only exception to this is when I make a milk & honey soap, I have to refridgerate the lye to cool it right down, otherwise my soap overheats & separates. (it also goes in the fridge overnight)
Generally, the cooler you have your oils and your lye, the slower it moves to trace, so if you are wanting time to play with colour, fragrance, etc, soaping with cooler oils & lye can be better.
Alternatively, you can soap warmer and hand stir to trace. I've been doing this lately, and find that it doesn't really take much longer, and I have more control over the whole process. I actually find thin trace instead of stickblending straight past it
So, to make a long answer short, I don't believe that the temp of your lye really makes a huge difference, generally. Try it and see