Now your last point is valid
stickN'rudder007, I had not factored that in. Yes, if the increased air density does lead to increased thrust, it will reduce Vmu.
On the other hand, at lower (typical) Takeoff altitudes and not so high temperatures (depending on the engine rating), if the engine is 'Flat Rated', there will be no difference in thrust. Most engines that I'm familiar with are flat rated up to about 30°C, so, below that it would make no difference, above that, Yup, it WILL make a difference.
Good thinking processes, you make a good aeronautical detective.
Regards,
Old Smokey