Last Plaque? Gloucester Arms

I doubt it!

I suppose it's a reasonable assumption that, if a Pub is called the Gloucester Arms, the not-quite-plaque over the door is said Duke's coat-of-Arms.

As has been said, "Many a time and oft" a Rose by any other name...

"Dockray Hall is a 15th century hall house, which has been absorbed in the construction of a 16th century coaching inn. Richard, duke of Gloucester, later King Richard III is reputed to have stayed here, when alterations were being made at the castle and the room where he slept, is now a store room. On the western edge of town is Penrith Castle and nearby is St Andrew's Church tower."

There are, apparently traces of the 15th century still visible inside.

Not that any of this looks 15th C.