  This work investigates Early Dark Energy (EDE) scenarios as a potential
precombination solution to the Hubble tension problem in the $F(R)$ gravity
theory. We first develop a dimensionless quantity to visualize the density
ratio between the EDE field and matter. Following existing scenarios, we then
discuss conditions under which the Hubble tension could be alleviated by
introducing a temporary injection of $10\%$ fractional energy around the time
of matter-radiation equality between $z=10^{3}-10^{4}$. We further confront
these models with local gravity tests and find inconsistencies with
observations. Taking a broader view, we convert constraints from local gravity
tests into constraints for general EDE in $F(R)$ gravity. We eventually arrive
at a no-go theorem for $F(R)$ gravity, which restricts nontrivial modifications
to General Relativity in the early universe.
