Why does additional term to electromagnetic Lagrangian leave Maxwell's equations unchanged?

Question

The addition of

$$\mathcal{L}' = \epsilon_{\mu\nu\rho\sigma}F^{\mu\nu}F^{\rho\sigma} \propto \vec{E}\cdot\vec{B}$$

to the electromagnetic Lagrangian density leaves Maxwell's equations unchanged (shown here).

In Carroll's GR book, he appends the question: "Can you think of a deep reason for this?". What might this be? I imagine there is a physical argument other than simply "gauge invariance".

Answer 1

As e.g. shown in OP's linked Phys.SE post the $F\wedge F$ term is a total divergence/topological term, and hence doesn't contribute to the Euler-Lagrange (EL) equations, i.e. Maxwell's equations, cf. e.g. this Phys.SE post.