The current-voltage characteristics in the charge order state of the two-dimensional organic conductor α-(BEDT-TTF)(2)I(3) exhibit power law behavior at low temperatures. The power law is understood in terms of the electric-field-dependent potential between electrons and holes, which are thermally excited from the charge order state. The power law exponent steeply changes from 1 to 3 in the range from 30 to 45 K with decreasing temperature, thereby suggesting the occurrence of a Kosterlitz-Thouless-type transition; many (few) unbound electron-hole pairs are thermally excited above (below) the transition. The effects of the finite size and interlayer coupling on the power law behavior are discussed.