We study the eigenvalues of the MOTS stability operator for the Kerr black hole with angular momentum per unit mass $|a| ll M$. We prove that each eigenvalue depends analytically on $a$ (in a neighbourhood of $a=0$), and compute its first nonvanishing derivative. Recalling that $a=0$ corresponds to the Schwarzschild solution, where each eigenvalue has multiplicity $2ell+1$, we find that this degeneracy is completely broken for nonzero $a$. In particular, for $0 < |a| ll M$ we obtain a cluster consisting of $ell$ distinct complex conjugate pairs and one real eigenvalue. As a special case of our results, we get a simple formula for the variation of the principal eigenvalue. For perturbations that preserve the total area or mass of the black hole, we find that the principal eigenvalue has a local maximum at $a=0$. However, there are other perturbations for which the principal eigenvalue has a local minimum at $a=0$.
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