  We show the temperature dependence such as smoothness and monotone
decreasingness with respect to the temperature of the solution to the
BCS-Bogoliubov gap equation for superconductivity. Here the temperature belongs
to the closed interval $[0,\, \tau]$ with $\tau>0$ nearly equal to half of the
transition temperature. We show that the solution is continuous with respect to
both the temperature and the energy, and that the solution is Lipschitz
continuous and monotone decreasing with respect to the temperature. Moreover,
we show that the solution is partially differentiable with respect to the
temperature twice and the second-order partial derivative is continuous with
respect to both the temperature and the energy, or that the solution is
approximated by such a smooth function.
