We studied the effect of varying gas concentration, buffer gas, length, and type of fibers on the performance of optical fiber photothermal phase modulators based on C2H2-filled hollow-core fibers. For the same control power level, the phase modulator with Ar as the buffer gas achieves the largest phase modulation. For a fixed length of hollow-core fiber, there exists an optimal C2H2 concentration that achieves the largest phase modulation. With a 23-cm-long anti-resonant hollow-core fiber filled with 12.5% C2H2 balanced with Ar, phase modulation of π-rad at 100 kHz is achieved with a control power of 200 mW. The modulation bandwidth of the phase modulator is 150 kHz. The modulation bandwidth is extended to ∼1.1 MHz with a photonic bandgap hollow-core fiber of the same length filled with the same gas mixture. The measured rise and fall time of the photonic bandgap hollow-core fiber phase modulator are 0.57 µs and 0.55 µs, respectively.