Our paper introduces a new theoretical framework called the Fractional
Einstein--Gauss--Bonnet scalar field cosmology, which has important physical
implications. Using fractional calculus to modify the gravitational action
integral, we derived a modified Friedmann equation and a modified Klein--Gordon
equation. Our research reveals non-trivial solutions associated with
exponential potential, exponential couplings to the Gauss--Bonnet term, and a
logarithmic scalar field, which are dependent on two cosmological parameters,
m and
α0=t0H0 and the fractional derivative order
μ. By
employing linear stability theory, we reveal the phase space structure and
analyze the dynamic effects of the Gauss--Bonnet couplings. The scaling
behavior at some equilibrium points reveals that the geometric corrections in
the coupling to the Gauss--Bonnet scalar can mimic the behavior of the dark
sector in modified gravity. Using data from cosmic chronometers, type Ia
supernovae, supermassive Black Hole Shadows, and strong gravitational lensing,
we estimated the values of
m and
α0, indicating that the solution
is consistent with an accelerated expansion at late times with the values
α0=1.38±0.05,
m=1.44±0.05, and
μ=1.48±0.17 (consistent
with
Ωm,0=0.311±0.016 and
h=0.712±0.007), resulting in an age
of the Universe
t0=19.0±0.7 [Gyr] at 1
σ CL. Ultimately, we
obtained late-time accelerating power-law solutions supported by the most
recent cosmological data, and we proposed an alternative explanation for the
origin of cosmic acceleration other than
ΛCDM. Our results generalize
and significantly improve previous achievements in the literature, highlighting
the practical implications of fractional calculus in cosmology.