We propose a subvolume method to study the $θ$ dependence of the free energy density of the four-dimensional SU($N$) Yang-Mills theory on the lattice. As an attempt, the method is first applied to SU(2) Yang-Mills theory at $T=1.2\,T_c$ to understand the systematics of the method. We then proceed to the calculation of the vacuum energy density and obtain the $θ$ dependence qualitatively different from the high temperature case. The numerical results combined with the theoretical requirements provide the evidence for the spontaneous CP violation at $θ= π$, which is in accordance with the large $N$ prediction and indicates that the similarity between 4d SU($N$) and 2d CP$^{N-1}$ theories does not hold for $N$=2.

6 pages, 8 figures, text clarified, references added, systematic errors estimated, argument on surface tension omitted, Fig.7 added to clarify the limitation of the method, no significant change in conclusion. published version