  The complete form of the high-temperature expansion of the one-loop
contribution to the free energy of a scalar field on a stationary gravitational
background is derived. The explicit expressions for the divergent and finite
parts of the high-temperature expansion in a three-dimensional space without
boundaries are obtained. These formulas generalize the known one for the
stationary spacetime. In particular, we confirm that for a massless conformal
scalar field the leading correction to the Planck law proportional to the
temperature squared turns out to be nonzero due to non-static nature of the
metric. The explicit expression for the so-called energy-time anomaly is found.
The interrelation between this anomaly and the conformal (trace) anomaly is
established. The natural simplest Lagrangian for the "Killing vector field" is
given.
