  We discuss the connection between the integer moments of the Fermi
distribution function that occur in the Sommerfeld expansion and the
coefficients that occur in anomalous conservation laws for chiral fermions. As
an illustration we extract the chiral magneto-thermal energy current from the
mixed gauge-gravity anomaly in the 3+1 dimensional energy-momentum conservation
law. We then use a similar method to confirm the conjecture that the $T^2/12$
thermal contribution to the chiral vortical effect (CVE) current arises from
the gravitational Pontryagin term in the 3+1-dimensional chiral anomaly
