Cosmos++: relativistic magnetohydrodynamics on unstructured grids with local adaptive refinement. A new code and methodology are introduced for solving the general relativistic magnetohydrodynamic (GRMHD) equations in fixed background spacetimes using time-explicit, finite-volume discretization. The code has options for solving the GRMHD equations using traditional artificial viscosity (AV) or nonoscillatory central difference (NOCD) methods, or a new extended AV (eAV) scheme using artificial viscosity together with a dual energy/flux-conserving formulation. The dual-energy approach allows for accurate modeling of highly relativistic flows at boost factors well beyond what has been achieved to date by standard artificial viscosity methods. It provides the benefit of Godunov methods in capturing high Lorentz boosted flows, but without complicated Riemann solvers, and the advantages of traditional artificial viscosity methods in their speed and flexibility. In addition, the GRMHD equations are solved on an unstructured grid that supports local adaptive mesh refinement using a fully threaded oct-tree (in three dimensions) network to traverse the grid hierarchy across levels and immediate neighbors. A number of tests are presented to demonstrate robustness of the numerical algorithms and adaptive mesh framework over a wide spectrum of problems, boosts, and astrophysical applications, including relativistic shock tubes, shock collisions, magnetosonic shocks, Alfvén wave propagation, blast waves, magnetized Bondi flow, and the magnetorotational instability in Kerr black hole spacetimes.Cosmos++: relativistic magnetohydrodynamics on unstructured grids with local adaptive refinement.