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The thermodynamic potential for superfluid $^3$He-B embedded in ahomogeneously distributed random potential is calculated from a quasiclassicalreduction of the Luttinger-Ward functional to leading order in $k_{\mbox{$\tinyB$}} T_c/E_f$. The resulting functional provides an extension of theGinzburg-Landau free energy functional to all temperatures $0<T\le T_c$.Theoretical predictions based on this functional for the heat capacity ofsuperfluid $^3$He-B embedded in homogeneous, isotropic silica aerogel are ingood agreement with experimental reports for superfluid $^3$He-B infused into98.2% porous silica aerogel over the pressure range $p=11 - 29\,\mbox{bar}$.The analysis supports a conclusion that superfluid $^3$He-B infused intohigh-porosity silica aerogels is a gapless superfluid at all pressures.

The Heat Capacity of $^3$He-B in Silica Aerogel