With my TR-6 the top only goes up if I get caught out in a rainstorm. (tonneau cover)
This isn't always a good idea though. While doing the testing for the tire test (coming in March CMS) I drove out to the test facility with the top down in 95 degree heat. (hour and a half drive) After taking a few laps with my helmet on, I nearly collapsed from the heat. The TR never seemed to notice the heat though, and ran beautifully.
Of course I'm not the sharpest knife in the drawer, so I drove home with the top down and the sun beating down on my head. It took me a few hours to recover once getting home. Not my smartest move, as I nearly gave myself a heat stroke!