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We report the effects on the thermodynamic properties of a 3D Bose gas causedby a temperature dependent energy gap $\Delta (T)$ at the lower edge of theenergy spectrum of the particles constituting the Bose gas which behaves likean ideal Bose gas when the gap is removed. Explicit formulae are given for thecritical temperature, the condensate fraction, the internal energy and theisochoric specific heat, which are calculated for three different gaps thatabruptly go to zero at temperature $T_B$, as well as for the dampedcounterparts whose drop to zero we have smoothed. In particular, for theundamped BCS (Bardeen, Cooper and Schrieffer) gap it is observed that theBose-Einstein condensation (BEC) critical temperature $T_c$ is equal to that ofthe ideal Bose gas $T_0$, for all $T_B \leq T_0$; surprisingly, the condensatefraction presents two different filling rates of the ground state at $T_c =T_0$ and at $T_B < T_0$; while the specific heat shows a finite jump at $T_c$as well as a divergence at $T_B$. Three-dimensional infinite Bose gas resultsare recovered when the temperature independent gap is either a constant orequal to zero.

Two-step BEC coming from a temperature dependent energy gap