  We study the thermodynamics in the BPS limit of AdS black holes realizing the
topological twist. We use a limiting procedure that allows us to reach the
extremal point along a trajectory in the space of supersymmetric Euclidean
solutions. We show that on this space we can write a quantum statistical
relation, which is well-defined in the BPS limit and relies on imposing a
suitable constraint among the chemical potentials, due to supersymmetry and
regularity. We stress the importance of this in relating the thermal partition
function of the dual field theory to the topologically twisted index.
