I've been using my rx 480 for about a month now (got the 4gb version) and i've noticed that all times the memory clock of the card is always at 1750mhz when my second screen is connected. From test that I've seen only the same goes for the 8gb version. My previous card was the nvidia 960 which didn't have this issue and would always clock back the memory clock when the second monitor was connected at idle state. What I find really odd is that with my nvidia 960 when I would turn my refresh rate from my primary monitor to 144hz the memory clock would increase in order to keep up. However the memory clocks of the rx r480 stays 300mhz when my main display is set at a 144hz refresh rate and there is no second monitor connected.
I can't imagne that when I am browsing that al all times the card needs maxed out memory speed for a second monitor, while at 144hz it's fine with 300mhz idle. I was looking into WattMan but there is no option to control the memory clock or force a state when idle on desktop for example. The only thing I've been able to is to overclock my memory. I really hope that in the future we can adjust the memory state in the driver or a new memory state will be added in an opcomming driver(if possible) for the memory which would allow for much less power consumption. My card is reaching quite high idle tempatures because of this whenever I got a second monitor connected.
Source for idle multi monitor consumption: MSI RX 480 Gaming X 8 GB Review | techPowerUp