I have a 2000 TL with apprioximately 115k. After short errrands if I start the car and don't hold the key long enough it will stumble, and run rough on start up. Twice now I had to crank it for maybe 10-15 seconds and it would catch, stumble, and then run ok. No check engine lights. I usually keep half full or more with 91 octane. Timing belt was done at 85K, I have not done a tune up.

 

Probably "Heatsoak", Search is your friend.

 

Indeed heatsoak or a vapor lock issue. If you turn it to on(without cranking it over) and pressing the accelerator 1 time does it start faster?