According to Gauss's Law, the total electric flux ($\phi$) through a closed surface is equal to $\frac{1}{\epsilon_0}$ times the enclosed charge ($q_{enclosed}$), i.e., $\phi = \oint \mathbf{E} \cdot d\mathbf{S} = \frac{q_{enclosed}}{\epsilon_0}$.

1. Calculate Electric Field at Surface: At the surface of the sphere of radius $r = a$, the magnitude of the electric field is $E = A(a) = Aa$.
2. Calculate Flux: Since the field is radially outward, it is parallel to the area vector at every point on the sphere's surface. The surface area of the sphere is $S = 4\pi a^2$. Flux $\phi = E \times S = (Aa) \times (4\pi a^2) = 4\pi A a^3$.
3. Calculate Charge: Using Gauss's Law, $q_{enclosed} = \epsilon_0 \times \phi = \epsilon_0 (4\pi A a^3) = 4\pi\epsilon_0 A a^3$.