I like the green also .
My concern is that stone cladding so close to the end of the OTA will cause worse issues (convection currents) than the other stone/brick/asphalt shingle and driveway that is part of the project - but farther away from the equipment.
I have researched this but can't come up with anything definitive. What I find are discussions about elevating an obs to overcome the "ground effect" of warm earth surrounding an obs. I have elevated this obs above the asphalt driveway and above the stone cladding near the ground and hope this will be helpful. So I worry that I might now "elevate the ground" up to the obs by installing the stone cladding at the obs level.
And I remind myself that I'm not just building an obs where I can give priority to astronomy in every decision. The project is a family vacation home that needs to make everybody happy. I just don't want to do something (else?) crazy dumb at the final stage of the project. This is literally the last exterior decision.
You can look up the specific heats of all the materials near the observatory, and roughly calculate how long it'd take for the daytime heat to be shed from the roof and nearby stone facing. My guess is that the small additional thermal mass of the small amount of stone would be a drop in the bucket compared to the massive roof.