  The tunneling density of states in dirty superconductors is often well
described by the phenomenological Dynes formula. Recently we have shown that
this formula can be derived, within the coherent potential approximation, for
superconductors with simultaneously present pair-conserving and pair-breaking
impurity scattering. Here we demonstrate that the theory of such so-called
Dynes superconductors is thermodynamically consistent. We calculate the
specific heat and critical field of the Dynes superconductors, and we show that
their gap parameter, specific heat, critical field, and penetration depth
exhibit power-law scaling with temperature in the low-temperature limit. We
also show that, in the vicinity of a coupling constant-controlled
superconductor to normal metal transition, the Homes law is replaced by a
different, pair-breaking dominated scaling law.
