  We show that the quasi-Euclidean sections of various rotating black holes in
different dimensions possess at least one non-conformal negative mode when
thermodynamic instabilities are expected. The boundary conditions of fixed
induced metric correspond to the partition function of the grand-canonical
ensemble. Indeed, in the asymptotically flat cases, we find that a negative
mode persists even if the specific heat at constant angular momenta is
positive, since the stability in this ensemble also requires the positivity of
the isothermal momentum of inertia. We focus in particular on Kerr black holes,
on Myers-Perry black holes in five and six dimensions, and on the Emparan-Reall
black ring solution. We go on further to consider the richer case of the
asymptotically AdS Kerr black hole in four dimensions, where thermodynamic
stability is expected for a large enough cosmological constant. The results are
consistent with previous findings in the non-rotation limit and support the use
of quasi-Euclidean instantons to construct gravitational partition functions.
