  The Euclidean path integral is compared to the thermal (canonical) partition
function in curved static space-times. It is shown that if spatial sections are
non-compact and there is no Killing horizon, the logarithms of these two
quantities differ only by a term proportional to the inverse temperature, that
arises from the vacuum energy. When spatial sections are bordered by Killing
horizons the Euclidean path integral is not equal to the thermal partition
function. It is shown that the expression for the Euclidean path integral
depends on which integral is taken first: over coordinates or over momenta. In
the first case the Euclidean path integral depends on the scattering phase
shift of the mode and it is UV diverge. In the second case it is the total
derivative and diverge on the horizon. Furthermore we demonstrate that there
are three different definitions of the energy, and the derivative with respect
to the inverse temperature of the Euclidean path integral does not give the
value of any of these three types of energy. We also propose the new method of
computation of the Euclidean path integral that gives the correct equality
between the Euclidean path integral and thermal partition function for
non-compact spaces with and without Killing horizon.
