Let me start off by saying there isn't a perfect time. For each profile or page it depends on your friends or audience.
According to Bit.ly, links sent between 1:00 p.m. and 4:00 p.m. get the most traction, with Wednesday at 3:00 p.m. being the best time to post on Facebook all week. Links posted after 8:00 p.m. and before 8:00 a.m. on Facebook don't get the most clicks. Bit.ly recommends not posting the links you want to go viral during the weekend. "While traffic starts to increase around 9:00 a.m., one would be wise to wait to post until 11am," bit.ly said in a blog post on its site. "Traffic from Facebook fades after 4:00 p.m."
But as someone that works in sports and entertainment the optimal times to post to Facebook are not the same. Between 11:00am and 4:00pm is good if you want to use that time to remind your audience of programming you have coming on later that night such as a pro-sports game or awards show. But obviously if that programming is occurring at night, you should also be posting while the show is on or right after. And the weekends are a peak time for sports content.
I say all that to say, really look at your fan base, be mindful of Edgerank and here are some tips and tricks from the experts on how to increase your fan engagement.