  The weak-coupling limits of the gap and critical temperature computed within
Eliashberg theory surprisingly deviate from the BCS theory predictions by a
factor of $1/\sqrt{e}$. Interestingly, however, the ratio of these two
quantities agrees for both theories. Motivated by this result, here we
investigate the weak-coupling thermodynamics of Eliashberg theory, with a
central focus on the free energy, specific heat, and the critical magnetic
field. In particular, we numerically calculate the difference between the
superconducting and normal-state specific heats, and we find that this quantity
differs from its BCS counterpart by a factor of $1/\sqrt{e}$, for all
temperatures below $T_{c}$. We find that the dimensionless ratio of the
specific-heat discontinuity to the normal-state specific heat reduces to the
BCS prediction given by $\Delta C_{V}(T_{c})/C_{V,n}(T_c)\approx1.43$. This
gives further evidence to the expectation that all dimensionless ratios tend to
their "universal values" in the weak-coupling limit.
