Gas-filled hollow-core photonic crystal fibers are used to stabilize a fiber laser to the 13C2H2 P(16) (ν1+ν3) transition at 1542 nm using saturated absorption. Four hollow-core fibers with different crystal structure are compared in terms of long term lock-point repeatability and fractional frequency instability. The locked fiber laser shows a fractional frequency instability below 4×10−12 for averaging time up to 104 s. The lock-point repeatability over more than 1 year is 1.3×10−11, corresponding to a standard deviation of 2.5 kHz. A complete experimental investigation of the light-matter interaction between the spatial modes excited in the fibers and the frequency of the locked laser is presented. A simple theoretical model that explains the interaction is also developed.