gap[t_]  :=  
      y /. FindRoot[ 
            y  ==  Tanh[y/t], {y, 0.5} ][[1]]

gap[ .5]

Plot[gap[t],  {t,  .001, 0.999},  
      PlotRange  -> { {0, 1},  {0, 1}},  
      PlotPoints  ->  100]