  We report the effects on the thermodynamic properties of a 3D Bose gas caused
by a temperature dependent energy gap $\Delta (T)$ at the lower edge of the
energy spectrum of the particles constituting the Bose gas which behaves like
an ideal Bose gas when the gap is removed. Explicit formulae are given for the
critical temperature, the condensate fraction, the internal energy and the
isochoric specific heat, which are calculated for three different gaps that
abruptly go to zero at temperature $T_B$, as well as for the damped
counterparts whose drop to zero we have smoothed. In particular, for the
undamped BCS (Bardeen, Cooper and Schrieffer) gap it is observed that the
Bose-Einstein condensation (BEC) critical temperature $T_c$ is equal to that of
the ideal Bose gas $T_0$, for all $T_B \leq T_0$; surprisingly, the condensate
fraction 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 results
are recovered when the temperature independent gap is either a constant or
equal to zero.
