(TiZrHf)₅₀Ni₃₀Cu_20-xCo_x (x=2, 4, 6, at%) high-entropy high-temperature shape memory alloys were fabricated by water-cooled copper crucible in a magnetic levitation vacuum melting furnace, and the effects of Co content on microstructure and mechanical properties were investigated. The results indicate that the grain size of the alloy decreases with increasing the Co content. In the as-cast state, the alloy consists primarily of the B19′ phase, with a trace of B2 phase. The fracture morphology is predominantly composed of the B19′ phase, whereas the B2 phase is nearly absent. Increasing the Co content or reducing the sample dimensions (d) markedly enhance the compressive strength and ductility of the alloy. When d=2 mm, the (TiZrHf)₅₀Ni₃₀Cu₁₄Co₆ alloy demonstrates the optimal mechanical properties, achieving a compressive strength of 2142.39±1.8 MPa and a plasticity of 17.31±0.3%. The compressive cyclic test shows that with increasing the compressive strain, the residual strain of the (TiZrHf)₅₀Ni₃₀Cu₁₄Co₆ alloy increases while the recovery ability declines. The superelastic recovery capability of the alloy is continuously enhanced. The superelastic recovery rate increases from 1.36% to 2.12%, the residual strain rate rises from 1.79% to 5.52%, the elastic recovery rate ascends from 3.86% to 7.36%, while the total recovery rate declines from 74.48% to 63.20%.