$$
\text { 7. } 4 \sqrt{2}-4 \text {. }
$$

Let the side length of the base of the quadrilateral pyramid be $a$. Then the distance from the center of the sphere to the base is $\sqrt{1-\frac{1}{2} a^{2}}$.
$$
\begin{array}{l}
\text { From } \frac{\sqrt{1-\frac{1}{2} a^{2}}}{\frac{a}{2}}=\frac{\frac{\sqrt{2}}{2} a}{1+\sqrt{1-\frac{1}{2} a^{2}}} \\
\Rightarrow a^{2}=4 \sqrt{2}-4,
\end{array}
$$

Thus, the area of the base of the quadrilateral pyramid is $4 \sqrt{2}-4$.
7. A regular quadrilateral pyramid $P-ABCD$ is circumscribed around a sphere with a radius of 1. If the distance from the center of the sphere to each face of the pyramid is equal, then the area of the base of the pyramid is $\qquad$ .