Why does the base-emitter voltage of a BJT decrease with temperature?
\$I_S \$ is highly temperature dependent.  As the temperature of the material increases, more electron-hole pairs are thermally generated, increasing \$I_S \$.   Here's a link that gives the formula SPICE uses for \$I_S \$


  Temperature appears explicitly in the exponential terms of the BJT and
  diode model equations. In addition, saturation currents have a
  built-in temperature dependence. The temperature dependence of the
  saturation current in the BJT models is determined by:


\$ I_S(T_1)=I_S(T_0)\left[\dfrac{T_1}{T_0}\right]^{XTI}exp\left[{\dfrac{-E_gq(T_1T_0)}{k(T_1-T_0)}}\right] \$

I believe that \$I_S \$ is roughly cubic in \$ T \$