PCM needs to see 180F before running in closed loop. If you are continuously running under 180, PCM will probably stay in open loop or bounce back and forth. This will normally cause it to run much richer, although I'm not 100% sure on that. These cars ran rich from the factory, but not sure what Justin did for that on the tune.
This is what I'd do:
#1 - change your oil sending/sensor unit - that's not something you want to mess around with.
#2 - while you're under there, change your oil to Mobil 1 5W-30, or the High Mileage variant if you're over 75k. Use an Delco PF13 oil filter - it has a much larger media exposure which means more oil flow through the filter, which in turn will give you better filtration.
#3 - as TenZen said, change your t-stat and cap. Stant makes a good cap, and try to get a t-stat made in USA, Canada, or Israel (yes, Israel). T-stat could be sticking open, or cap could be bleeding off pressure.
#4 - change your coolant temp sensor
#5 - reset PCM with a 30 minute battery disconnect
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'05 Stone White - a few mods...