How to prove this matrix inequality $\det(A+B)\ge 2^n\sqrt{\det(A)\det(B)}$

This seems to be a bit late for an answer, but the following theorem is proven in "Linear Algebra" by Lax in chapter 10.

Let $A$ and $B$ be self-adjoint, positive, $n \times n$ matrices. Then for all $0<t<1,$ \begin{align} \det(tA + (1-t)B) \geq (\det A)^{t}(\det B)^{1-t}. \end{align} Your answer follows with $t = \frac{1}{2}$.

This is a corollary of Minkowski's Determinant Theorem: $\det(A+B)^\frac{1}{n}\geq \det(A)^\frac{1}{n}+\det(B)^\frac{1}{n}.$ Apply AM-GM inequality to the right-hand side.