A new approach to conquer the thermal phase drift of an optical fiber Fabry-Perot interferometer (FPI) sensor is proposed and experimentally demonstrated. By employing a hollow-core anti-resonant fiber (HC-ARF) and optimizing the fusion splicing (includes mode field adaptation) between the lead-in single-mode fiber (SMF) and the HC-ARF, a high spectral resolution (λ/Δλ ≈ 3.8 × 104) optical fiber air-cavity FPI sensor with a fringe visibility higher than 7 dB is constructed. To eliminate the thermal phase drift (i.e. temperature crosstalk) of the sensor that originates from the intrinsic thermal expansion effect of the silica material of the HC-ARF, the FPI air cavity is connected to the external environments, by which the effect of air expelling from the cavity with temperature increasing can well compensate the temperature-induced cavity elongation. As a result, the thermal phase drift of the FPI is reduced to zero at a temperature range of ∼ 80-110 °C and within the temperature range of 40-80 °C, the thermal phase drift is still halved compared with the sealed FPI cavity. The nearly zero thermal phase drift of a FPI at such a temperature range has never been achieved before, to our best knowledge. As a proof of concept, a temperature-immune fiber-optic strain sensor is demonstrated. This work offers a new and efficient approach to eliminate the thermal phase drift (i.e. temperature crosstalk) of a fiber-optic device, which may significantly improve the measurement accuracy and detection limit of fiber-optic FPI sensors. Furthermore, the principle and schema can be generalized to a wide variety of fiber-optic devices.