  The role of the equation of state for a perfectly conducting, relativistic
magnetized fluid is the main subject of this work. The ideal constant
$\Gamma$-law equation of state, commonly adopted in a wide range of
astrophysical applications, is compared with a more realistic equation of state
that better approximates the single-specie relativistic gas. The paper focus on
three different topics. First, the influence of a more realistic equation of
state on the propagation of fast magneto-sonic shocks is investigated. This
calls into question the validity of the constant $\Gamma$-law equation of state
in problems where the temperature of the gas substantially changes across
hydromagnetic waves. Second, we present a new inversion scheme to recover
primitive variables (such as rest-mass density and pressure) from conservative
ones that allows for a general equation of state and avoids catastrophic
numerical cancellations in the non-relativistic and ultrarelativistic limits.
Finally, selected numerical tests of astrophysical relevance (including
magnetized accretion flows around Kerr black holes) are compared using
different equations of state. Our main conclusion is that the choice of a
realistic equation of state can considerably bear upon the solution when
transitions from cold to hot gas (or viceversa) are present. Under these
circumstances, a polytropic equation of state can significantly endanger the
solution.
