What could cause a peak to split into 2 resonances as temperature increases in NMR spectroscopy
Since I do not know which compound it is, my guess is that you have two different  $\ce{H^1}$ that have almost the same chemical shift at room temperature. The chemical shifts change with temperature as for instance in sucrose:



Here  are represented mainly the  $\ce{H^1}$  of the $\ce{-OH}$ but the $\ce{C-H}$ (the peak of 1) shifts also.  

In your case,  one is almost fixed, while the other one moves to higher $\delta$  and, therefore, they split into two. 

More info on the sucrose spectrum here.