  The interstellar medium of galaxies is governed by supersonic turbulence,
which likely controls the star formation rate (SFR) and the initial mass
function (IMF). Interstellar turbulence is non-universal, with a wide range of
Mach numbers, magnetic fields strengths, and driving mechanisms. Although some
of these parameters were explored, most previous works assumed that the gas is
isothermal. However, we know that cold molecular clouds form out of the warm
atomic medium, with the gas passing through chemical and thermodynamic phases
that are not isothermal. Here we determine the role of temperature variations
by modelling non-isothermal turbulence with a polytropic equation of state
(EOS), where pressure and temperature are functions of gas density,
P~rho^Gamma, T~rho^(Gamma-1). We use grid resolutions of 2048^3 cells and
compare polytropic exponents Gamma=0.7 (soft EOS), Gamma=1 (isothermal EOS),
and Gamma=5/3 (stiff EOS). We find a complex network of non-isothermal
filaments with more small-scale fragmentation occurring for Gamma<1, while
Gamma>1 smoothes out density contrasts. The density probability distribution
function (PDF) is significantly affected by temperature variations, with a
power-law tail developing at low densities for Gamma>1. In contrast, the PDF
becomes closer to a lognormal distribution for Gamma<=1. We derive and test a
new density variance - Mach number relation that takes Gamma into account. This
new relation is relevant for theoretical models of the SFR and IMF, because it
determines the dense gas mass fraction of a cloud, from which stars form. We
derive the SFR as a function of Gamma and find that it decreases by a factor of
~5 from Gamma=0.7 to Gamma=5/3.
