For applications such as laser cooling of doped fibers, where it is critical to measure accurately the temperature of a cooled fiber that is very close to room temperature, it is paramount to develop a reliable, very short (mm) fiber temperature sensor with millikelvin resolution and very little drift. We report a second generation of a unique slow-light fiber Bragg grating (FBG) temperature sensor that meets these stringent requirements. Experiments and modeling establish that its temperature response depends only on well-known material constants of the silica FBG and the response of the probe laser's wavelength controller. The response is independent of the linewidth of the slow-light resonance, hence different FBGs and/or resonances have the exact same response. Examples of measured cooling in optically pumped Yb-doped fibers show that more reliable thermal contact with the cooled fiber is obtained by wrapping the FBG and the cooled fiber.