I'm 35 and it definitely takes me longer to get moving. Im my younger days I would sometimes clock the first mile to make sure I was running 6:00 pace or better and then just hang on to that pace for the rest of the run. Now I start off most runs at a shuffle and it takes me a couple of miles before I even think about doing anything fast. I probably have more endurance than I did at 20 and can run faster over ling distances, but it just takes me longer to get moving.
I'm probably a bit smarter about training too. JK's ideas about starting off laughably slow and "letting the pace come to you" make a lot of sense to me. This approach has helped me increase my mileage to its highest level ever without injury at age 35.
I've also experimented with progression runs as warmups for workouts. I might start off at a shuffle and slowly work up to tempo pace over about 20 minutes. I use this warmup for intervals and tempo workouts. I'll usually run the lastminute or two at tempo pace or faster then take a break to stretch. When I first get up to that pace, it feels fast but after the stretching break it makes it much easier to run fast during the workout itself.