Graphene-based hyperbolic metamaterials provide a unique scaffold for designing nanophotonic devices with active functionalities. In this work, we have theoretically demonstrated that the characteristics of a polarization-dependent tunable hyperbolic microcavity in the mid-infrared frequencies could be realized by modulating the thickness of the dielectric layers, and thus breaking periodicity in a graphene-based hyperbolic metamaterial stack. Transmission of the tunable microcavity shows a Fabry-Perot resonant mode with a Q-factor > 20, and a sixfold local enhancement of electric field intensity. It was found that by varying the gating voltage of graphene from 2 to 8 V, the device could be self-regulated with respect to both the intensity (up to 30%) and spectrum (up to 2.1 µm). In addition, the switching of the device was considered over a wide range of incident angles for both the transverse electric and transverse magnetic modes. Finally, numerical analysis indicated that a topological transition between elliptic and type II hyperbolic dispersion could be actively switched. The proposed scheme represents a remarkably versatile platform for the mid-infrared wave manipulation and may find applications in many multi-functional architectures, including ultra-sensitive filters, low-threshold lasers, and photonic chips.