  In this work we investigate the effects of a geometrically generated early
dark energy era on the energy spectrum of the primordial gravitational waves.
The early dark energy era, which we choose it to have a constant equation of
state parameter $w$, is synergistically generated by an appropriate $f(R)$
gravity in the presence of matter and radiation perfect fluids. As we
demonstrate, the predicted signal for the energy spectrum of the $f(R)$
primordial gravitational waves is amplified and can be detectable, for various
reheating temperatures, especially for large reheating temperatures. The signal
amplitude depends on the duration of the early dark energy era and on the value
of the dark energy equation of state parameter, with the most latter affecting
more crucially the amplification. Specifically the amplification occurs when
the equation of state parameter approaches the de Sitter value $w=-1$.
Regarding the duration of the early dark energy era, we find that the largest
amplification occurs when the early dark energy era commences at a temperature
$T=0.85\,$eV until $T=7.8\,$eV. Moreover we study a similar scenario in which
amplification occurs, where the early dark energy era commences at $T=0.29\,$eV
and lasts until the temperature is increased by $\Delta T\sim 1.7\,$eV.
