In our experience once our Paulownia got to be seven or eight years old, many of the leaves browned and went black and some of the main stems also died back. I put that down to drought here but it may also be due to poor growing conditions as our heavy clay soil is not liked by many trees.
By cutting your Paulownia right down very low near the ground in winter, you can rejuvenate it, as it will grow new stems next spring and these tend to have huge leaves to make a very decorative plant. You can in fact do this every year to keep it as an herbacious shrub rather than a tree, especially as it has difficulty keeping the flower buds in place over winter for flowering the next spring in a British climate.
Paulownias tend to be short lived so maybe that is why your leaves are going black, or it could be due to excessive rain.
By digging up a good section of young root near the base in spring, they are easy to grow from root cuttings, so you could also have another plant as a back up.

Thank you Bertie, a very comprehensive answer, much appreciated.