Should, I would think. When I had my Tundra, the non-tow package had this, and it was thermostatically controlled. On mine, I had the tow package, with a big cooler, and there was still a t-stat. For fun one day I stuck a pin in, as required for doing fluid changes, just to see what would happen if I forced it into using the cooler "always". Warmed up much slower, but I don't recall if it impacted torque convertor lockup or not. Decided to leave it alone, while not lifetime the old advice about fluid temperature don't apply like they used to.
On mine I could only access (on my Scangauge) the convertor outlet temperature. I'm not sure if the temperature spike there should be feared or not. But yeah, I could see 230+ with stupid ease in that truck on hills with the TC unlocked. Sad thing is, I swear, I could get better mpg if I forced a downshift before the hill and then carried it with the TC locked...