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