  We employ the isomonodromic method to study linear scalar massive
perturbations of Kerr black holes for generic scalar masses $M\mu$ and generic
black hole spins $a/M$. A degeneracy between the longest-living quasinormal
mode and the first overtone is found for $(M\mu)_c \simeq 0.3704981$ and
$(a/M)_c\simeq 0.9994660$. We show that the longest-living mode and the first
overtone change continuously into each other as we vary the parameters around
the point of degeneracy, providing evidence for the existence of a geometric
phase around an exceptional point. We interpret our findings through a
thermodynamic analogy.
