These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Is Emi a fanatic who takes her frustrations out on Maou because her violent Knight Templar tendencies don't fly in our world, or is her expectations of Maou perfectly justified after everything he did to her people, putting her in an impossible situation now that he's become harmless and sympathetic and her sense of justice won't allow her to just end the threat anymore?
Emi's Jerk Ass attitude towards Maou irks many viewers. Others find it hilarious and justified, given that he was pretty much fantasy Hitler in their world and is threatening to return and try to subjugate them all over again.
Episode 7 worsens her status. Is Emi punching and kneeing Maou hilarious or did it show how much of a bitch she is to Maou? Take note that Maou saved her from falling to the ground.
She's generally hated on /a/
Is Chiho hilarious and heartwarming? Or is she annoying and controlling? Some fans think it's the first, while others think it's the latter.
Some "fans" seem to vehemently insist that the inclusion of an Harem nature into the anime is dragging it down, despite there not actually being a Harem and only one girl openly in love with Maou. Anything to do with Emi and the other girl concerning Maou are only misunderstandings, results of flawed cultural osmosis or not what they seem to be.
The whole series is this to those fans under the misconception that it's supposed to be High Fantasy and that the Slice of Life elements are hampering it down.
Conversely, when the light novels' narrative gradually shifted its focus to the High Fantasy elements of the story(while still keeping its signature humor), there are also fans who complain that the "innovative" Slice of Life humor had been ditched in favor of the "generic" LN fantasy fare.
The Demon Lord and his most loyal general live together in a cramped, run-down apartment in which the former does the working and the latter does the housework. Oh my. Alciel actually calls himself a "house husband" in his bio, although he acts more like an overprotective parent to Maou at times. Lampshaded by Chiho's classmate who goes wild with yaoi fantasies upon learning Alciel and Maou are roomies.
Arguably worse now that Lucifer is living with them. - "...boneheads..."