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The Euclidean path integral is compared to the thermal (canonical) partitionfunction in curved static space-times. It is shown that if spatial sections arenon-compact and there is no Killing horizon, the logarithms of these twoquantities differ only by a term proportional to the inverse temperature, thatarises from the vacuum energy. When spatial sections are bordered by Killinghorizons the Euclidean path integral is not equal to the thermal partitionfunction. It is shown that the expression for the Euclidean path integraldepends on which integral is taken first: over coordinates or over momenta. Inthe first case the Euclidean path integral depends on the scattering phaseshift of the mode and it is UV diverge. In the second case it is the totalderivative and diverge on the horizon. Furthermore we demonstrate that thereare three different definitions of the energy, and the derivative with respectto the inverse temperature of the Euclidean path integral does not give thevalue of any of these three types of energy. We also propose the new method ofcomputation of the Euclidean path integral that gives the correct equalitybetween the Euclidean path integral and thermal partition function fornon-compact spaces with and without Killing horizon.

Is the Euclidean path integral always equal to the thermal partition function?