A scheme for measuring a small variation rate of the refractive index (RI) based on orbital angular momentum (OAM) interferometry and time-frequency analysis is proposed and demonstrated in this paper. Two vortex beams carrying OAM of opposite signs are used for interference to produce a petal-like intensity distribution. The variation in RI of the sample leads to a time variable phase delay between the reference and measurement paths, and causes the rotation of the petal-like spots. The rotation angular velocity of the petal-like spots is proportional to the RI variation rate, the normalized cross correlation method is used to estimate it. Then, a time-frequency analysis method is employed to study the time evolution of the variation rate of RI. Three kinds of RI models with different variation rates are simulated and the results are consistent with expectations. The proposed measurement method is simple in structure, and extends a new approach to detect other physical coefficients of RI or the tiny velocity.