Doubles your time? No. At best it is 1.5 times, but it doesn't work on everything. My mission 64 target time was 17m13s the first time I did it. I beat it in just under 13 minutes... I got 3 stars.
However, I got the gold watch and equipped it and then tried again. I beat him again and this time it said my target time was 17m09s. Apparently it does not affect mission time (at least on 64). So I tried a few normal enemies and what I was pretty sure used to be 52s was around 1m20s now.
Take that as you will.
"We must educate our friends. For without education, our friends are just... stupid people that we know." - George of the Jungle