Sometimes people want to be too smart. If you want to consume a handful different restful API, it might make sense to abstract away some common functionality in your client implementation — yet to assume they follow the same convention in how their URI is built is borderline insane.

All I wanted to do was to change one API to a newer version, and now the implementation breaks for at least two other because it was done in an Abstract class and now I have to untangle that mess.

In some cases code duplication wouldn't be that bad. Even if an otherwise unrelated API seemingly share the same contract, still assume it has its own contract. You never know how those API evolve and I proclaim they will evolve towards breaking your assumptions.