In this paper, a strain-insensitive temperature sensor based on a dual polarization fiber grating laser is demonstrated. The laser is fabricated by inscribing two wavelength-matched Bragg gratings in an Er-doped fiber. It emits single-longitudinal-mode output in wavelength domain and generates a RF-domain signal as a beat note between the two polarization modes. A temperature sensor has been exploited by monitoring the beat frequency. The measured temperature sensitivity is -78.46 kHz/°C. Theoretical analysis suggests that the temperature response is a result of both the differences in thermo-optic coefficient and thermal expansion between the core and cladding. In contrast, the sensor is almost insensitive to applied axial strain. We found that the strain insensitivity is due to the compensation between the strain-induced birefringence change and the effect of the elongation/material index change. The proposed sensor can be applied for reliable and precise measurement of temperature independently, towards the applications in structural integrity, oil-well monitoring, aerospace engineering, and process control.