Wild box turtles live outside year round.

Unless you are well outside your Three-toed's natural range, you should be able to house the turtle outside, assuming that you provide it with an adequate habitat which includes various options for shade, sun, a water source, cover, etc.
To give you an example, my Easterns live outside here in Central Virginia, where it gets plenty hot (over 100 on occasion), but they have plenty of shady areas in their enclosure as well as a small pond where they can beat the heat.

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