  Using the anomaly inflow mechanism, we compute the flavor/Lorentz
non-invariant contribution to the partition function in a background with a
U(1) isometry. This contribution is a local functional of the background
fields. By identifying the U(1) isometry with Euclidean time we obtain a
contribution of the anomaly to the thermodynamic partition function from which
hydrostatic correlators can be efficiently computed. Our result is in line
with, and an extension of, previous studies on the role of anomalies in a
hydrodynamic setting. Along the way we find simplified expressions for
Bardeen-Zumino polynomials and various transgression formulae
