For a massless free scalar field in a globally hyperbolic space-time we compare the global temperature $$T=\beta ^{-1}$$ T = β - 1 , defined for the $$\beta $$ β -KMS states $$\omega ^{(\beta )}$$ ω ( β ) , with the local temperature $$T_{\omega }(x)$$ T ω ( x ) introduced by Buchholz and Schlemmer. We prove the following claims: (1) whenever $$T_{\omega ^{(\beta )}}(x)$$ T ω ( β ) ( x ) is defined, it is a continuous, monotonically decreasing function of $$\beta $$ β at every point x. (2) $$T_{\omega }(x)$$ T ω ( x ) is defined when M is ultra-static with compact Cauchy surface and non-trivial scalar curvature $$R\ge 0$$ R ≥ 0 , $$\omega $$ ω is stationary, and a few other assumptions are satisfied. Our proof of (2) relies on the positive mass theorem. We discuss the necessity of its assumptions, providing counter-examples in an ultra-static space-time with non-compact Cauchy surface and $$R<0$$ R < 0 somewhere. Our results suggest that under suitable circumstances (in particular in the absence of acceleration, rotation and violations of the weak energy condition in the background space-time) both notions of temperature provide qualitatively similar information, and hence the Wick square can be used as a local thermometer.