  We discuss the Bekenstein generalized entropy of Schwarzschild black hole,
with the contribution of an external matter field affected by degrees of
freedom inside the event horizon. To take into accountthis effect to the
generalized entropy, we use Euclidean functional methods. In the Euclidean
section of the Schwarzschild manifold, we consider an Euclidean quantum
effective model, a scalar theory in the presence of an additive quenched
disorder. The average the Gibbs free energy over the ensemble of possible
configurations of the disorder is obtained by the distributional zeta-function
method. In the series representation for the average free energy with
respective effective actions emerges the generalized Schr\"{o}dinger operators
on Riemannian manifolds. Finally, is presented the generalized entropy density
with the contributions of the black hole geometric entropy and the external
matter fields affected by the internal degrees of freedom. The validity of the
generalized second law using Euclidean functional methods is obtained.
