We show a unique pressure effect on the electronic structure of a layered organic solid beta'-(BEDT-TTF)(2)AuCl2 by a theoretical study based on density functional theory. At ambient pressure, atomic and electronic structures of this compound are very similar to those of beta'-(BEDT-TTF)(2)ICl2, which shows superconducting transitions at high pressures (p > 7 GPa). Contrary to the expectation that the title compound may also show a superconducting behavior at high pressures, it is recently reported that it shows neither superconducting nor metallic behaviors even when the pressure up to 9.9 GPa is applied. For the future attempts to discover new physical properties of layered organic solids by applying extremely high pressures, it is very important to understand what kind of differences exist between the pressure effects on these two similar salts. In this study, we clarify the qualitative differences between the pressure effects on the electronic structures of these two salts. We show that the electronic structure of the title compound at high pressures is very unique among those of layered organic solids. The width of the band crossing the Fermi level of the title compound does not increase after the pressure higher than 8 GPa is applied, and its electronic structure at high pressures shows three-dimensional rather than quasi-one- or two-dimensional characteristics. We have found that this unique feature comes from the fact that the AuCl2 part directly contributes to the electronic structure near the Fermi level.