I own a 2011 XJL, that has 99,700 miles on it. I had to replace the engine at 82,000 miles due to the engine getting so hot it melted the fuel lines and warped the block and head. I finally reveived settlement from the Insurance to pay for the new engine. The temperature guage never registered beyond the middle mark on the indicator.

I now have 17,000 on the new engine and was driving about 45 minutes at between 5 and 15 MPH when it was 90 outside. This time my temperature guage did show hot and an warning light came on showing hot. I am not sure what causes this nor can the dealer replicate it either. My theory is that the fan is not large enough to draw the appropriate amount of air into the radiator to keep it cooled down while traveling at slow speeds.

 

Wow, driving for 45 minutes at only 5-15 mph would heat up any car, given the extended duration and higher revs generated by using the low gears for so long.

I suspect you may be right...the car is just unable to cool the engine down when driven at such low speeds for so long.