  We consider the standard model of i.i.d. bond percolation on $\mathbb Z^d$ of
parameter $p$. When $p>p_c$, there exists almost surely a unique infinite
cluster $\mathcal C_p$. Using the recent techniques of Cerf and Dembin, we
prove that the variance of the graph distance in $\mathcal C_p$ between two
points of $\mathcal C_p$ is sublinear. The main result extends the works of
Benjamini, Kalai and Schramm, Benaim and Rossignol and Damron, Hanson and Sosoe
for the study of the variance of passage times in first passage percolation
without moment conditions on the edge-weight distribution.
