Why do NMR coupling constants change with temperature?
Let’s use vicinal 3-bond $\ce{H-C-C-H}$ proton coupling constants as an example.  Coupling is a through bond phenomenon and as such the coupling constant will change depending if we have better (the coupling constant will increase) or worse (the coupling constant will decrease) overlap between the two $\ce{C-H}$ bonds involved.  Overlap is maximized when the two, vicinal $\ce{C-H}$ bonds are aligned in an eclipsed conformation ($\ce{H-C-C-H }$ dihedral angle =0°) or anti ($\ce{H-C-C-H }$ dihedral angle =180°)  conformation.  When the dihedral angle is ~90° the overlap of the two vicinal $\ce{C-H}$ bonds is minimized.  This thinking is embodied in the Karplus equation which relates vicinal coupling constants to dihedral angle.  The following graph illustrates this dependence of the vicinal coupling constant on the dihedral angle.



image source

All of this tells us that an experimentally observed vicinal coupling constant will be dependent upon the relative populations of the eclipsed, staggered and anti conformations. Said differently, the vicinal coupling constant will be an average of each of the individual coupling constants for each conformation, weighted by the relative population of that conformation. Since the relative conformer populations change with temperature, the coupling constant will also change with temperature.