The influence of Hf element on enthalpy and hardness relaxation behavior of Zr₇₀Al_7.5Ni₈Cu_14.5 (70Zr), Zr₅₅Al₁₀Ni₅Cu₃₀ (55Zr) and (Zr_0.75Hf_0.25)₆₅Al_7.5Ni₁₀Cu_17.5 (65Zr_0.75Hf_0.25) bulk glass-type alloys with large diameters of centimeter-level was examined by measuring the temperature dependence of apparent specific heats in as-spun and annealed states. Results show that the structural relaxation of 70Zr and 55Zr alloys occurs through a single-stage mode where the relaxation peak is observed at the annealing temperature (T_a) near glass transition temperature (T_g), while 65Zr_0.75Hf_0.25 alloy shows a double-stage reaction with enthalpy relaxation peaks appearing at 523 and 648 K near T_g. The single-stage relaxation mode for 70Zr and 55Zr alloys indicates that these alloys have high resistance to annealing-induced relaxation over the whole temperature range up to T_g, because the distinct relaxation occurs only around the temperature of T_g. 65Zr_0.75Hf_0.25 glassy alloy shows a relaxation sub-peak around 523 K presumably due to the weaker bonding nature of Zr-Hf atomic pair with nearly zero in heat of mixing, and the distinct main relaxation occurs near T_g. The Vickers hardness also shows the similar double-stage peak behavior as a function of T_a for 65Zr_0.75Hf_0.25 as well as (Zr_0.5Hf_0.5)₆₅-Al_7.5Ni₁₀Cu_17.5 (65Zr_0.5Hf_0.5) alloys, so the T_a for the first-stage hardness peak agrees with that for the enthalpy relaxation. These results imply that icosahedral-like medium range ordered structure consisting of Zr, Al, Ni and Cu elements with the three components rule for stabilization of supercooled liquid remains stable during the low T_a relaxation. The double-stage enthalpy and hardness relaxation only appear in 65Zr_0.75Hf_0.25 and 65Zr_0.5Hf_0.5 glassy alloys which deviate from the three components rule, implying that the unnecessary multiplication for the formation of bulk glassy alloys leads to an increase in structural instability during low-temperature annealing.