When the data range in matplotlib contourf does not correspond to the range of colorbar, matplotlib automatically adjust the range, even when set_clim is set manually.
(In other words, I needed a fixed colorbar regardless of the data)
The solution is set "levels" as follows;
interval = np.linspace(2.5,4.5, num=32, endpoint=True)
img=plt.contourf(LON, LAT, DATA,
vmin=2.5,vmax=4.5,
cmap = "jet",
levels=interval,
transform=ccrs.PlateCarree())
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