I use Clockwork Tomato for android and it does exactly what you ask in the expansion pack. The free version is great with a simple interface and keeps a log of how many pomodoros and the time you took on each pomodoro (in case you extend it or stop early) that can be accessed by swiping the screen. It keeps a pretty extensive log so far as I can tell going back a long time.
Or, if you need/want to export the logs or have task integration, then you can pay $2.50 for the expansion pack. In the expansion pack, the logs become more detailed (based on which project you are working on), you can have additional profiles, and you can import/export the logs and tasklist to Dropbox. They have a free 10-day trial of the expansion which showed me that I don't need it, because the free version is the best app I have found for my needs, but if you need the export functionality this might be a great option for you.
With 'app' you mean software on a phone? Any platform in particular? – THelper – 2012-12-13T08:33:17.160
Android would be preferable, but on a computer would be okay too. – John Salvatier – 2012-12-13T20:05:44.140
Technically, Pomodroido stores the state (aborted/completed) and timestamps of every pomodoro you do with the app. But it doesn't display it anywhere. – Muz – 2012-12-20T03:58:20.870
Hmm, that's pretty interesting, how do you know? Would it be easy to hack it to store that somewhere? – John Salvatier – 2012-12-20T08:37:09.650