diff --git a/xarray/core/groupby.py b/xarray/core/groupby.py
index f47de953abc..83fa8b5c138 100644
--- a/xarray/core/groupby.py
+++ b/xarray/core/groupby.py
@@ -101,29 +101,24 @@ def __init__(self, obj, group, squeeze=False, grouper=None):
         """
         from .dataset import as_dataset
 
+        self._stacked_dim = None
+        group_name = group.name
         if group.ndim != 1:
             # try to stack the dims of the group into a single dim
             # TODO: figure out how to exclude dimensions from the stacking
             #       (e.g. group over space dims but leave time dim intact)
             orig_dims = group.dims
             stacked_dim_name = 'stacked_' + '_'.join(orig_dims)
-            # the copy is necessary here
+            # the copy is necessary here, otherwise read only array raises error
+            # in pandas: https://github.com/pydata/pandas/issues/12813
+            # Is there a performance penalty for calling copy?
             group = group.stack(**{stacked_dim_name: orig_dims}).copy()
-            # without it, an error is raised deep in pandas
-            ########################
-            # xarray/core/groupby.py
-            # ---> 31     inverse, values = pd.factorize(ar, sort=True)
-            # pandas/core/algorithms.pyc in factorize(values, sort, order, na_sentinel, size_hint)
-            # --> 196     labels = table.get_labels(vals, uniques, 0, na_sentinel, True)
-            # pandas/hashtable.pyx in pandas.hashtable.Float64HashTable.get_labels (pandas/hashtable.c:10302)()
-            # pandas/hashtable.so in View.MemoryView.memoryview_cwrapper (pandas/hashtable.c:29882)()
-            # pandas/hashtable.so in View.MemoryView.memoryview.__cinit__ (pandas/hashtable.c:26251)()
-            # ValueError: buffer source array is read-only
-            #######################
-            # seems related to
-            # https://github.com/pydata/pandas/issues/10043
-            # https://github.com/pydata/pandas/pull/10070
             obj = obj.stack(**{stacked_dim_name: orig_dims})
+            self._stacked_dim = stacked_dim_name
+            self._unstacked_dims = orig_dims
+            # we also need to rename the group name to avoid a conflict when
+            # concatenating
+            group_name += '_groups'
         if getattr(group, 'name', None) is None:
             raise ValueError('`group` must have a name')
         if not hasattr(group, 'dims'):
@@ -157,7 +152,7 @@ def __init__(self, obj, group, squeeze=False, grouper=None):
             unique_coord = Coordinate(group.name, first_items.index)
         elif group.name in obj.dims:
             # assume that group already has sorted, unique values
-            if group.dims != (group.name,):
+            if group.dims != (group_name,):
                 raise ValueError('`group` is required to be a coordinate if '
                                  '`group.name` is a dimension in `obj`')
             group_indices = np.arange(group.size)
@@ -169,7 +164,7 @@ def __init__(self, obj, group, squeeze=False, grouper=None):
         else:
             # look through group to find the unique values
             unique_values, group_indices = unique_value_groups(group)
-            unique_coord = Coordinate(group.name, unique_values)
+            unique_coord = Coordinate(group_name, unique_values)
 
         self.obj = obj
         self.group = group
@@ -249,6 +244,13 @@ def _maybe_restore_empty_groups(self, combined):
             combined = combined.reindex(**indexers)
         return combined
 
+    def _maybe_unstack_array(self, arr):
+        """This gets called if we are applying on an array with a
+        multidimensional group."""
+        if self._stacked_dim is not None and self._stacked_dim in arr.dims:
+            arr = arr.unstack(self._stacked_dim)
+        return arr
+
     def fillna(self, value):
         """Fill missing values in this object by group.
 
@@ -399,7 +401,9 @@ def apply(self, func, shortcut=False, **kwargs):
             grouped = self._iter_grouped_shortcut()
         else:
             grouped = self._iter_grouped()
-        applied = (maybe_wrap_array(arr, func(arr, **kwargs)) for arr in grouped)
+        applied = (self._maybe_unstack_array(
+                    maybe_wrap_array(arr,func(arr, **kwargs)))
+                    for arr in grouped)
         combined = self._concat(applied, shortcut=shortcut)
         result = self._maybe_restore_empty_groups(combined)
         return result
diff --git a/xarray/test/test_dataarray.py b/xarray/test/test_dataarray.py
index d0f83457e68..aa118dec663 100644
--- a/xarray/test/test_dataarray.py
+++ b/xarray/test/test_dataarray.py
@@ -1250,8 +1250,10 @@ def test_groupby_multidim(self):
                             'lat': (['ny','nx'], [[10,10],[20,20]] ),},
                     dims=['ny','nx'])
         for dim, expected_sum in [
-                ('lon', DataArray([0, 3, 3], coords={'lon': [30,40,50]})),
-                ('lat', DataArray([1,5], coords={'lat': [10,20]}))]:
+                ('lon', DataArray([0, 3, 3],
+                               coords={'lon_groups': [30,40,50]})),
+                ('lat', DataArray([1,5],
+                               coords={'lat_groups': [10,20]}))]:
             actual_sum = array.groupby(dim).sum()
             self.assertDataArrayIdentical(expected_sum, actual_sum)