I've done a bit of poking around on the internet, because I had this question, too.

Much of what I've read says that as long as you sealed the container with an O2 absorber, it should be fine.

I was still a little concerned until I realized that brown sugar is nothing but white sugar and molasses. Why not pick up two bottles of molasses (one bottle of blackstrap for dark brown sugar and one bottle of regular) and plan on mixing it up when you need it? 1 cup of white sugar + 1 Tbsp of molasses. That would help you avoid problems.