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Metamaterials with active tunability of electromagnetic response have attracted much attention due to their flexibility and applicability in a wide range of optoelectronic applications. In this study, a mid-infrared (MIR) H-shaped metamaterial absorber (HMA) with high tunability is proposed. The structure of HMA is composed of one longitudinal and two transverse metal bars, forming an H-shaped resonator. By varying structural parameters, the HMA achieves perfect triple-band absorption. The resonant intensities of first and second resonances can be modulated linearly at the wavelength of 5.3 μm, while the resonant intensity of third resonance can be kept as stable at the wavelength of 7.3 μm. By moving the transverse metal bar, the maximum tuning range of resonance is 290 nm. Due to the non-rotationally symmetric configuration of HMA, it exhibits polarization-dependence characteristic. HMA exhibits the switchable function from triple-resonance to double-resonance. These results prove that the proposed HMA can be used for applications in gas sensing, polarization controlling, and IR filtering fields.