Actually you move slightly faster than a person that isn't wearing a stat hat.
When a person is using Beast mode gold, they have no movement penalty at all (so if they carry a heavy object, they'll move at the regular running speed), but if you combine a stat hat too, you move slightly faster than the regular running speed, which is kinda funny.
That is actually a common misconception.
With gold you walk a tiny bit slower than somebody who isn't carrying anything.
The stat hat eliminates that and you have the same speed as somebody carrying nothing.
Well I'm not sure about THAT one, but Gold + stat hat definitely make you move faster than normal running speed. Trust me, I've used BM in Co-op a lot when I play 3 player Co-op instead of 2 and I've noticed how players are falling behind when I'm carrying the idol.
This felt like it started happening since the medal change for objectives, but it felt like the overall movement was decreased for BM gold + hat. I mean usually you booked it with these, but now it just feels less fast.