The Unruh effect is one of the most startling predictions of quantum field theory. Its interpretation has been controversially discussed, since the first publications of Fulling, Davies and Unruh in the 1970ties. Buchholz and Solveen in 2013 proposed an application of basic thermodynamic definitions to clarify the meaning of temperature and thermal equilibrium in the Unruh effect. As a result the interpretation of the KMS parameter as an expression of local temperature has been questioned. Recently, Verch and Buchholz established that KMS states of a massless scaler field cannot be distinguished in remote regions of the Rindler spacetime, further questioning the thermal interpretation of the KMS parameter. My main result is to extend this asymptotic equivalence of KMS states to the case of a massive scalar field. The key argument involves establishing local quasi-equivalence of the disputed states in regions that infinitely extend towards an accelerated observers direction of motion.