  The aim of this paper is to find out a correspondence between one-loop
effective action $W_E$ defined by means of path integral in Euclidean gravity
and the free energy $F$ obtained by summation over the modes. The analysis is
given for quantum fields on stationary space-times of a general form. For such
problems a convenient procedure of a "Wick rotation" from Euclidean to
Lorentzian theory becomes quite non-trivial implying transition from one real
section of a complexified space-time manifold to another. We formulate
conditions under which $F$ and $W_E$ can be connected and establish an explicit
relation of these functionals. Our results are based on the Kaluza-Klein method
which enables one to reduce the problem on a stationary space-time to
equivalent problem on a static space-time in the presence of a gauge
connection. As a by-product, we discover relation between the asymptotic
heat-kernel coefficients of elliptic operators on a $D$ dimensional stationary
space-times and the heat-kernel coefficients of a $D-1$ dimensional elliptic
operators with an Abelian gauge connection.
