During the summer I have an extension on my outside faucet--it is basically a few foot section of hose with a faucet at the end where I connect my garden hose. Therefore that exterior faucet is on all the time and the water flow is turned off at the end of the section of hose. In the fall when I turn off the exterior faucet, there is a loud "Thurrump" in that interior line when the powder room toilet is flushed. The powder room faucet which is attached to the same water line does not cause this sound. This has only started to occur the past few years and was not evident for the first few years. Any ideas as to what might be causing this noise and how to correct it? Thanks!!

When does the sound start? When the toilet starts to refill or when its done refilling and turns off? It might just be getting worn and not shutting off fast, but slow(which might cause the valve to sputter) and causing the thumping sound.

The sound starts as the toilet tank stops filling, but the sound is not in the toilet itself but in the water line in the basement. This branch of the water line has the toilet and sink from the powder room on it and then the outside faucet at the end. This branch is only about 10 feet long.

Uour outside hose is acting like a water hammer breaker device. It takes up the flow of the water when its shut off fast at the toilet.
It eitehr sounds like the toilet valve is beginnging to fail or your water pressure is to high in the house. or a combo of both.
Most likly its the valve like Adam B said is starting to wear.