There is a growing consensus that in the present universe most baryons reside in galaxy clusters and groups in the form of highly ionized gas at temperatures of 10(6)-10(8) K. Th, H-like and He-like ions of the heavy elements can produce absorption features-the so-called "X-ray forest"-in the X-ray spectrum of a background quasar. We investigate the distribution of the X-ray absorption lines produced by this gas under three different cosmological models: the standard cold dark matter model with Omega(0) = 1, a flat model with Omega(0) = 0.3, and an open model with Omega(0) = 0.3. We give a semianalytic calculation of the X-ray forest distribution based on the Press-Schechter formalism, following Perna & Loeb (1998). We choose three ions (O VIII, Si XIV, and Fe XXV) and calculate the distribution functions, the number of absorbers along the line of sight (LOS) to a distant quasar versus redshift, and column density in a given ion. We find significant differences in the evolution of the distribution functions among the three cosmological models. Using Monte Carlo simulations, we simulate the distribution of X-ray absorption lines for 10,000 random LOS. We find that there are at least several O vm lines with column density higher than 10(16) cm(-2). Finally, we explore the possibility of detecting the X-ray forest with current and upcoming X-ray missions, and we present an XMM Reflection Grating Spectrometer (RGS) simulation of a representative quasar X-ray spectrum.