  We consider the third order Benjamin-Ono equation on the torus $\partial_t u=
\partial_x \left( -\partial_{xx}u-\frac{3}{2}u H\partial_x u -
\frac{3}{2}H(u\partial_x u) + u^3 \right).$ We prove that for any
$t\in\mathbb{R}$, the flow map continuously extends to $H^s_{r,0}(\mathbb{T})$
if $s\geq 0$, but does not admit a continuous extension to
$H^{-s}_{r,0}(\mathbb{T})$ if $0<s<\frac{1}{2}$. Moreover, we show that the
extension is not weakly sequentially continuous in $L^2_{r,0}(\mathbb{T})$. We
then classify the traveling wave solutions for the third order Benjamin-Ono
equation in $L^2_{r,0}(\mathbb{T})$ and study their orbital stability.
