Temps as high as 260 degrees F (126 C) should be OK. If it gets at or over 290 F (143 C) I would begin to get concerned, as there might be a bearing or two or three failing.
I would not be concerned about lower temps in the diffs, unless you are running in very cold ambient temps, then you should use something a little lower in viscosity, so it will get to where it needs to be.
The front diff on my truck runs as high as 250 F (121 C) or so in the summer, the rear diff gets to about 200 F (93 C), maybe 210. In the winter running through North Dakota, or up into Winnipeg it barely got to 220/170 F (104/77 C).
And synthetic lubes are better, especially when running at the extreme end of the load envelope. 80-90 gear lube (it is not oil) is the standard to use today, although I think they have something a little thinner for the rigs that run up in Yellowknife and the like in the winter, and something a little heavier for rigs running in the desert oilfields in the Middle East.
I pointed out that gear lube is not "oil" because it is graphite based, not petroleum based like motor oil. And unless you get it hot enough (like way hotter than anything you will see under normal circumstances), gear lube will not burn, where motor oil will.
By the way BYP, excellent idea to use a meat thermometer to check the diff sender. And the lube levels should be full enough that when you remove the fill plug the lube just dribbles out.