Is it ethical to present a second paper at a conference when the first paper and travel was funded by a separate organization? Here is the scenario (hypothetical at the moment):

I have two papers on entirely unrelated topics, but are both presentable at the same conference. Paper 1 was funded through funding agency A, but paper 2 was not. The conference has multiple sessions covering a wide variety of topics. My funding to travel to the conference was provided by funding agency A in order to present paper 1.

Is it ethical to present paper 1 and paper 2 at this conference (given that I pay the abstract fee for paper 2 out of pocket), or am I confined to presenting results from the paper that I was funded to present?

Note: This isn't a dilemma I face now, but one a bit further down the line.
I can't see any ethical issues at all. You seem to be properly fulfilling the obligations of your funding. What else you do at a conference is your own concern. The funders are willing to support you and you are doing what is required for the funding. Rest easy. 

I can't imagine that any funding agency would suggest (or even think) that you can't do other things at a conference. They certainly expect you to take advantage of the opportunities it affords.