It seems to be a common understanding at present that, once event horizons are in thermal equilibrium, the entropy-area law holds inevitably. However no rigorous verification is given to such a very strong universality of the law in multi-horizon spacetimes. Then, based on thermodynamically consistent and rigorous discussion, this paper suggests an evidence of breakdown of entropy-area law for horizons in Schwarzschild-de Sitter spacetime, in which the temperatures of the horizons are different. The outline is as follows: We construct carefully "two thermal equilibrium systems" individually for black hole event horizon (BEH) and cosmological event horizon (CEH), for which the Euclidean action method is applicable. The integration constant (subtraction term) in Euclidean action is determined with referring to Schwarzschild and de Sitter canonical ensembles. The free energies of the two thermal systems are functions of three independent state variables, and we find a similarity of our two thermal systems with the magnetized gas in laboratory, which gives us a physical understanding of the necessity of three independent state variables. Then, via the thermodynamic consistency with three independent state variables, the breakdown of entropy-area law for CEH is suggested. The validity of the law for BEH can not be judged, but we clarify the key issue for BEH's entropy. Finally we make comments which may suggest the breakdown of entropy-area law for BEH, and also propose two discussions; one of them is on the quantum statistics of underlying quantum gravity, and another is on the SdS black hole evaporation from the point of view of non-equilibrium thermodynamics.