  In the low density intergalactic medium (IGM) that gives rise to the
Lyman-alpha forest, gas temperature and density are tightly correlated. The
velocity scale of thermal broadening and the Hubble flow across the gas Jeans
scale are of similar magnitude (Hlambda_J ~ sigma_th). To separate the effects
of gas pressure support and thermal broadening on the Lya forest, we compare
spectra extracted from two smoothed particle hydrodynamics (SPH) simulations
evolved with different photoionization heating rates (and thus different Jeans
scales), imposing different temperature-density relations on the evolved
particle distributions. The turnover scales in the flux power spectrum and flux
autocorrelation function are determined mainly by thermal broadening rather
than pressure. However, the insensitivity to pressure arises partly from a
cancellation effect with a sloped temperature-density relation (T ~ rho^{0.6}
in our simulations): the high density peaks in the colder, lower pressure
simulation are less smoothed by pressure support than in the hotter simulation,
and it is this higher density gas that experiences the strongest thermal
broadening. Changes in thermal broadening and pressure support have comparably
important effects on the flux probability distribution (PDF), which responds
directly to the gas overdensity distribution rather than the scale on which it
is smooth. Tests on a lower resolution simulation show that our statistical
results are converged even at this lower resolution. While thermal broadening
generally dominates the longitudinal structure in the Lya forest, we show in
Paper II that pressure support determines the transverse coherence of the
forest observed towards close quasar pairs. [ABRIDGED]
