We present an interesting scheme of ultra-broadband perfect terahertz (THz) absorber with periodic-conductivity graphene metasurface, instead of the common metasurfaces of geometrically patterned graphene with uniform conductivity. By exploiting periodic modulation of graphene conductivity, the graphene metasurface absorber produces a sequence of dense high-order resonance modes with near-perfect absorption peaks. The density of resonant modes can be largely enhanced to perform an ultra-broadband even continuum absorption in THz regime with suppression of the modulation minimum of graphene conductivity. Under the resistive loss effect of graphene on the modes broadening, the THz perfect absorber can perform a record ultra-broadband of 3.71 THz, which covers more than one-third of the whole THz region of 0.1–10 THz. Such scheme of periodic-conductivity graphene metasurface is of interest in pushing forward to broadband THz absorber, which has promising application prospects in THz stealth, imaging and communication fields.