We report an open-cavity optical fiber Fabry-Pérot interferometer (FPI) capable of measuring refractive index with very low temperature cross-sensitivity. The FPI was constructed by splicing a thin piece of C-shaped fiber between two standard single-mode fibers. The refractive index (RI) response of the FPI was characterized using water-ethanol mixtures with RI in the range of 1.33 to 1.36. The RI sensitivity was measured to be 1368 nm/RIU at the wavelength of 1600 nm with good linearity. Thanks to its all-glass structure, the FPI exhibits very low temperature cross-sensitivity of 3.04 × 10⁻⁷ RIU/°C. The effects of cavity length on the performance of the sensor were also studied. A shorter cavity gives rise to broader measurement range while offering larger detection limit, and vice versa. What's more, the effect of material dispersion of analyte on the sensitivity of open-cavity FPIs was identified for the first time. The sensor is compact in size and easy to fabricate. It is potentially useful for label-free optical sensing of chemical and biological samples.