Recently, metamaterials have attracted considerable attention because of their unique properties and capability of being used in many areas of science. Among these applications, metamaterial absorber is the one researchers show much interests. On the basis of its electromagnetic responses to other material parameters, the metamaterial absorber can be applied to sensing. In this paper, a metamaterial absorber with an I-shaped unit cell is proposed and its favorable sensing characteristics in terahertz frequency range are numerically simulated in terms of frequency-domain algorithm. Influences of the thickness of the sample to be tested and the thickness of dielectric spacer of the sensing of metamaterial absorber on the frequency sensitivity, amplitude sensitivity, and the figure of merit of the refractive index, are studied in detail. Research results indicate that as the refractive index of the sample, whose thickness being fixed, increases, the resonant frequency red-shifts and the reflected amplitude increases. And when the thickness of the sample with a particular refractive index increases, the resonant frequency red-shifts and the reflected amplitude increases correspondingly. The above researches indicate that the sensing of thickness or refractive index of the sample to be tested (abbreviated as specimen) can be realized in a metamaterial absorber. The frequency sensitivity of the refractive index can reach 153.17 GHz/RIU and the amplitude sensitivity of the refractive index can reach 41.37%/RIU when the thickness of the sample is fixed at 40 μm. The frequency sensitivity of the refractive index increases as the thickness of the sample tested increases, but the increasing range gradually decreases. In addition, the amplitude sensitivity of the refractive index increases linearly with the increase of thickness of the sample tested. The frequency sensitivity of thickness decreases linearly with the increase of the thickness of the sample to be tested which is of a particular refractive index. As the thickness of dielectric spacer increases, the frequency sensitivity of the refractive index increases until the thickness reaches 30 μm. Besides, when the refractive index takes a particular value, the frequency sensitivity of thickness decreases linearly as the thickness of dielectric spacer increases. Along with the gradual increase of the thickness of the sample tested, RFOM increases but the increasing range decreases. And TFOM gradually decreases as the thickness of sample tested increases. Both the RFOM and TFOM decrease with the increase of the thickness of dielectric spacer. In the end, the sensing mechanism of metamaterial absorber is discussed in detail. The reflectance spectra and the sensitivity can be adjusted with changing the refractive index and thickeness of the sample tested and the thickness of dielectric spacer, and this will provide important instructive means for terahertz sensing with metamaterial absorbers.