This will be possible when our Historics product becomes available over the next few weeks.
Our REST API buffers Tweets for a short amount of time from the live stream of the filter you set up.
Alternatively, consistently running a stream to collect these results may not be as expensive as you think:
A simple stream like the one below looks for mentions of "Apple" (a very popular subject on Twitter), but only collects the Tweets if they have been shared or retweeted a certain number of times. This brings your costs down by removing any Tweets which were never retweeted from your stream, and only tracks Tweets that are popular, and have been retweeted several times.
The stream itself costs $4.80 to run for 24 hours + a license fee for the Tweets which comes in at $0.10 per 1000 Tweets (at time of writing). This stream seems to return less than 10 Tweets per minute, so an estimate for license costs for the day would only be an extra $0.70 - $1.40.