  We provide a transparent discussion of the high temperature asymptotic
behaviour of Cosmology in a dilaton-Einstein-Gauss-Bonnet (dEGB) scenario of
modified gravity with vanishing scalar potential. In particular, we show that
it has a clear interpretation in terms of only three attractors (stable
critical points) of a set of autonomous differential equations:
$w=-\frac{1}{3}$, $w=1$ and $1<w<\frac{7}{3}$, where $w\equiv p/\rho$ is the
equation of state, defined as the ratio of the total pressure and the total
energy density. All the possible different high-temperature evolution histories
of the model are exhausted by only eight paths in the flow of the set of the
autonomous differential equations. Our discussion clearly explains why five out
of them are characterized by a swift transition of the system toward the
attractor, while the remaining three show a more convoluted evolution, where
the system follows a meta-stable equation of state at intermediate temperatures
before eventually jumping to the real attractor at higher temperatures.
Compared to standard Cosmology, the regions of the dEGB parameter space with
$w=-\frac{1}{3}$ show a strong enhancement of the expected Gravitational Wave
stochastic background produced by the primordial plasma of relativistic
particles of the Standard Model. This is due to the very peculiar fact that
dEGB allows to have an epoch when the energy density $\rho_{\rm rad}$ of the
relativistic plasma dominates the energy of the Universe while at the same time
the rate of dilution with $T$ of the total energy density is slower than what
usually expected during radiation dominance. This allows to use the bound from
BBN to put in dEGB a constraint $T_{\rm RH}\lesssim 10^8 - 10^9$ GeV on the
reheating temperature of the Universe $T_{\rm RH}$. Such BBN bound is
complementary to late-time constraints from compact binary mergers.
