  The phase diagram on the $\theta$-$T$ plane in four dimensional SU(3)
Yang-Mills theory is explored. We revisit the $\theta$ dependence of the
deconfinement transition temperature, $T_c(\theta)$, on the lattice through the
constraint effective potential for Polyakov loop. The $\theta$ term is
introduced by the reweighting method, and the critical $\beta$ is determined to
$\theta \sim 0.75$, where the interpolation in $\beta$ is carried out by the
multipoint reweighting method. The $\theta$ dependence of $T_c$ obtained here
turns out to be consistent with the previous result by D'Elia and Negro
\cite{DElia:2012pvq,DElia:2013uaf}. We also derive $T_c(\theta)$ by classifying
configurations into the high and low temperature phases and applying the
Clausius-Clapeyron equation. It is found that the potential barrier in the
double well potential at $T_c(\theta)$ becomes higher with $\theta$, which
suggests that the first order transition continues robustly above $\theta \sim
0.75$. Using information obtained here, we try to depict the expected $\theta$
dependence of the free energy density at $T < T_c(0)$, which crosses the first
order transition line at an intermediate value of $\theta$. Finally, how the
Lee-Yang zeros associated with the spontaneous CP violation appear is discussed
formally in the large $N$ limit, and the locations of them are found to be
$(\theta_R,\theta_I)=\left( (2m+1)\pi, \frac{2n+1}{2\chi V_4} \right)$ with $n$
and $m$ arbitrary integers.
