The thermodynamic analysis of the coexistence curves with any number of critical points in binary mixtures at constant pressure was carried out in the framework of the regular-mixing theory. We proposed a universal temperature dependence of the general energy interaction parameter both for one and for several critical points. An evolution of multicritical point coexistence curves was analyzed and a unified consideration of the boundaries of liquid-liquid and vapor-liquid equilibria was fulfilled. As a result, new properties of the multicritical point coexistence curves were derived and new phase transitions were theoretically predicted. The relationships between conjugate phase separation temperatures were applied to calculate a pressure dependence of mutual solubility of the 2-butanol/water mixtures as a system with three critical points. Moreover, the generalized reduced temperature for the systems with several critical points was introduced to compare scaling and regular-mixing approaches to the quantitative description of the published experimental data on coexistence curves with two critical points.