  Mutual information serves as an important measure of correlation between
subsystem components. In the framework of quantum field theories (QFTs) they
have better regulated UV behavior than entanglement entropy, and thus provide
more direct access to universal aspects of entanglement structures. In this
paper, we study the linear responses under shape deformation of the mutual
information in the conformal field theory (CFT) vacuum between two spheres of
radius $R$ separated by large distance $L\gg R$ or conformally equivalent
configurations. Our calculations make use of the previous OPE results for
mutual information \cite{Faulkner2016Aug} and the associated modular
Hamiltonian \cite{Faulkner2021Aug}. In particular, we apply the entanglement
first law to compute the linear responses of mutual information under shape
deformation on one of the spheres. We find that the linear responses exhibit a
high degree of universality for a selected class of OPE contributions. We
demonstrate that there is a "little group" of symmetries associated with the
set-up. Our result implies that the spherical mutual information is extremal
over shape deformations of non-zero modes under the symmetry group.
