The rules for multiplying costs and weight to get the additional cost of mithral apply only to weapons and non-armor items.
It is perfectly reasonable to say that the price of raw mithral stays the same regardless of it's intended use. That said, the price of various mithral items is influenced by other factors, not only the price of raw material, which is why the disparate pricing seems to occur. Looking at it the other way, you would never use 50 lbs of mithril to make a 25 lbs of resulting armor. Maybe some small amount on top of 25 lbs would get wasted, which is unlikely as usually metals are easily recycled. Even so, the weight of a mithril armor equals to not only weight of mithril plates themselves, but also leather straps, various kinds of fastenings such as wires or chains, gap-oprotecting chainmails, under-armor padding etc.
To complete your example, I would say that for an additional cost of 9000 gp you get 18 lbs of raw mithral, slightly less when you consider a masterworking fee of 150 gp included in the price. Subtracting the 18 lbs from 25 lbs of total armor weight, we are left with 7 lbs. It seems reasonable that for a full plate weighing 50 lbs, 7 lbs would be accounted to padding, fastenings, decoration etc. which is not lightened by using mithral. The price of 500 gp/lb holds.
Now moving on to weapons and other non-armor items, the cost of working mithral is calculated by taking the original weight. DuckTapeAl already established that mithral is a defensive material, so maybe it is the case that effective price doubling is related to additional effort put into shaping, hardening and sharpening a slab of mithral into a weapon. Also, with weapons proper balancing is paramount, so while making a sword from mithral, you would actually have to use lighter (and probably more expensive) hilt material, additional weights (preferably also mithral) to keep it up to speed. It's even worse with blunt weapons, as a warhammer needs to be tip-heavy for sufficient momentum, forcing the smith to re-engineer hilt diameter, length and weight to account for the lighter head. On top of that mithral is traditionally said as being very hard to work, which would force the smith to use more sophisticated tools and materials, such as better furnace fuel (for more heat), different hammers (that would not break when striking), unusual tempering fluids (gentler or more volatile) and more resistant sharpening stones (perhaps diamond dust or something similar). All in all that could possibly double the price to procure a specific weapon, even if indeed you need less raw mithral, again making the 500 gp/lbs rule hold.