Large density fluctuations developing near the phase transition point of the binary mixture affect physical parameters directly related to the electrokinetic coupling coefficient. Here the first electrokinetic measurements for a porous rock sample are carried out with a critical binary mixture of triethylamine-water, especially around the phase transition point. From the measured streaming potential coefficient, the zeta potential is obtained for the critical composition. The results show that there is no anomaly in the streaming potential coefficient as the temperature approaches the demixing temperature. It is also seen that the streaming potential coefficient and the zeta potential in magnitude decreases with increasing temperature. This observation is opposite to what has been observed in literature. It means that the properties of the electric double layer for the mixtures are different from those for aqueous electrolytes. Additionally, the zeta potential for the critical composition is predicted to fluctuate around the critical point.