  We discuss the transient and steady state fluctuation relation for a
mechanical system in contact with two deterministic thermostats at different
temperatures. The system is a modified Lorentz gas in which the fixed
scatterers exchange energy with the gas of particles, and the thermostats are
modelled by two Nos\'e-Hoover thermostats applied at the boundaries of the
system. The transient fluctuation relation, which holds only for a precise
choice of the initial ensemble, is verified at all times, as expected. Times
longer than the mesoscopic scale, needed for local equilibrium to be settled,
are required if a different initial ensemble is considered. This shows how the
transient fluctuation relation asymptotically leads to the steady state
relation when, as explicitly checked in our systems, the condition found in
[D.J. Searles, {\em et al.}, J. Stat. Phys. 128, 1337 (2007)], for the validity
of the steady state fluctuation relation, is verified. For the steady state
fluctuations of the phase space contraction rate $\zL$ and of the dissipation
function $\zW$, a similar relaxation regime at shorter averaging times is
found. The quantity $\zW$ satisfies with good accuracy the fluctuation relation
for times larger than the mesoscopic time scale; the quantity $\zL$ appears to
begin a monotonic convergence after such times. This is consistent with the
fact that $\zW$ and $\zL$ differ by a total time derivative, and that the tails
of the probability distribution function of $\zL$ are Gaussian.
