Treadmill vs. Road Running

Most likely explanation:

The treadmill at your local gym has not been calibrated in a long time.

Also old treadmills may not be able to maintain a constant speed while you are running on them.

Conclusion:

You cannot go by the speed on the treadmill, you should go by effort.

Caveats:

Perceived effort and heart rate may be elevated when running on a treadmill because of overheating. Using a fan in your face may help.

Treadmills are boring and uninteresting, and completely nothing like real running. They do work like magic with hill sprints, drills, and hill training though.

I also suspect that while running on a treadmill we tend to shorten our stride and thus potentially have to increase the turnover at a given pace. Only a theroy.