I sit in mine too for 45 minutes. The car switches itself on and off several times and after 45 mins I switch off and leave the car until next time.
Whilst I am in the car I pay no attention to the temperature gauge, but often will watch the Hybrid Battery being charged and discharged.
Mine has 62,000 miles on the clock.
Trust in the technology and have the car serviced by Lexus in accordance with the manufacturer`s schedule Fenn.