  It is known that the anomalous Chern-Simons (CS) coupling of O$_p$-plane is
not consistent with the T-duality transformations. Compatibility of this
coupling with the T-duality requires the inclusion of couplings involving one
R-R field strength. In this paper we find such couplings at order $\alpha'^2$.
  By requiring the R-R and NS-NS gauge invariances, we first find all
independent couplings at order $\alpha'^2$. There are $1,\, 6,\,28,\,20,\,
19,\, 2$ couplings corresponding to the R-R field strengths $F^{(p-4)}$,
$\,F^{(p-2)}$, $\,F^{(p)}$, $\,F^{(p+2)}$, $\,F^{(p+4)}$ and $F^{(p+6)}$,
respectively. We then impose the T-duality constraint on these couplings and on
the CS coupling $C^{(p-3)}\wedge R\wedge R$ at order $\alpha'^2$ to fix their
corresponding coefficients. The T-duality constraint fixes all coefficients in
terms of the CS coefficient. They are fully consistent with the partial
couplings that have been already found in the literature by the S-matrix
method.
