I am drawing custom shape boundary map which focuses on the Pacific Sector of Southern Ocean (160E~180~60W,-60S~-90S). When adding gridlines, the region (180~60W) lacks gridlines.

I have tried many solutions (mostly for rectangular projection), but failed.
import matplotlib.pyplot as plt
import matplotlib.path as mpath
import cartopy
import cartopy.crs as ccrs
import numpy as np
proj = ccrs.Stereographic(central_longitude=228, central_latitude=-70)
fig = plt.figure(figsize=(10,10))
ax = plt.axes([0,0,1,1],projection=proj)
latmin = -90
latmax = -60
lonmin = 150
lonmax = 305
lats = np.linspace(latmax, latmin, latmax - latmin + 1)
lons = np.linspace(lonmin, lonmax, lonmax - lonmin + 1)
vertices = [(lon, latmin) for lon in range(lonmin, lonmax + 1, 1)] + \
[(lon, latmax) for lon in range(lonmax, lonmin - 1, -1)]
boundary = mpath.Path(vertices)
ax.set_boundary(boundary, transform=ccrs.PlateCarree())
ax.set_extent([170, 320, -90, -60], ccrs.PlateCarree())
ax.add_feature(cartopy.feature.LAND, zorder=1, edgecolor='k')
ax.gridlines(draw_labels=True)
plt.show()

Gridlinesplotting is problematic in this particular case. The plot involves special boundary, the parallels crossing the international dateline -- these may be the causes of the problem. I hope the answer by @Rutger_Kassies is what you need. However, if you still need the original plot extent, here is another approach.