There's an easy solution that could bring instant trust to Facebook's trending news box, simply release what is algorithmically trending.
Facebook doesn't have to stop their employees from curating the news stories because there is value in that. Facebook wants variety in the news box and they want to make sure that true big front page type stories hit the trending box early. Most users certainly understand this.
However, using curators does not preclude them from providing a link in the box to a Trending Page showing users what is truly trending by the numbers. It should be in real-time and could even show the top 100 stories with full stats that are important in their trending algorithm. Stats shown should include number of links, clicks, likes, shares and post views in the last 15, 30, 60, 120 minutes.
An API to this data should be made available and an API to what their curators chose as trending and in what order should also be created. Developers and others, such as conservative groups which felt that Facebook has been biased, can create apps to monitor the trends posted on the Facebook front page versus the true unbiased Facebook algorithmically created trends in order to keep Facebook and its curation staff honest. Facebook employees would have to include a conservative news trend if their own algorithm is showing it be trending in a huge way!
So Facebook, get open, get honest and quickly solve your Trending Topics credibility gap by releasing real-time stats of what is algorithmically trending.