  The thermodynamic potential for superfluid $^3$He-B embedded in a
homogeneously distributed random potential is calculated from a quasiclassical
reduction of the Luttinger-Ward functional to leading order in $k_{\mbox{$\tiny
B$}} T_c/E_f$. The resulting functional provides an extension of the
Ginzburg-Landau free energy functional to all temperatures $0<T\le T_c$.
Theoretical predictions based on this functional for the heat capacity of
superfluid $^3$He-B embedded in homogeneous, isotropic silica aerogel are in
good agreement with experimental reports for superfluid $^3$He-B infused into
98.2% porous silica aerogel over the pressure range $p=11 - 29\,\mbox{bar}$.
The analysis supports a conclusion that superfluid $^3$He-B infused into
high-porosity silica aerogels is a gapless superfluid at all pressures.
