A universal feature of charged entanglement entropy

R\'enyi entropies, $S_n$, admit a natural generalization in the presence of global symmetries. These "charged R\'enyi entropies" are functions of the chemical potential $\mu$ conjugate to the charge contained in the entangling region and reduce to the usual notions as $\mu\rightarrow 0$. For $n=1$, this provides a notion of charged entanglement entropy. In this letter we prove that for a general $d (\geq 3)$-dimensional CFT, the leading correction to the uncharged entanglement entropy across a spherical entangling surface is quadratic in the chemical potential, positive definite, and universally controlled (up to fixed $d$-dependent constants) by the coefficients $C_J$ and $a_2$. These fully characterize, for a given theory, the current correlators $\langle JJ\rangle $ and $\langle TJJ \rangle$, as well as the energy flux measured at infinity produced by the insertion of the current operator. Our result is motivated by analytic holographic calculations for a special class of higher-curvature gravities coupled to a $(d-2)$-form in general dimensions as well as for free-fields in $d=4$. A proof for general theories and dimensions follows from previously known universal identities involving the magnetic response of twist operators introduced in arXiv:1310.4180 and basic thermodynamic relations.