The nature of this attack does require you to be targeting a specific piece of code (probably a library crypto function), however the manual processing would be no different if that code was running on a different service. The issue is that having a key is useless if you do not know what the key is for. You could probably use the same type of attack to get that information, but it would likely involve a great deal more manual labor because the data is not in a standard format. Also, given how cheep it is to run in parrallel with a target VM, it seems like it is more economical to pick a target anyway.

On an unrelated note, isn't the attack described in the paper not a timing attack because it gets the information by reading the cache instead of reading the clock?

I guess the same attack could theoretically work for an L2 cache, though you'd need an exponentially larger amount of data collection to pull it off, with an even bigger margin of error?

If you mean the paper linked as "slightly older research" (http://rump2009.cr.yp.to/8d9cebc9ad358331fcde611bf45f735d.pd...), it's slides from a talk entitled "Cross-VM Vulnerabilities in Cloud Computing" by Thomas Ristenpart, Eran Tromer, Stefan Savage, and Hovav Shacham, presented at Crypto'09's "rump session" (a session for short, amusing and/or late-breaking talks). Downloads slowly for me too, but eventually finishes (6mb pdf).

The following paper by the same authors, from a few months later, appears to be the published-paper version of that research, though, which is probably more useful: http://pages.cs.wisc.edu/~rist/papers/cloudsec.html