I don't know how anyone could keep their home at 80 degrees F during the summer, but as already well-covered by members above, a 5-degree variation in ambient-temperature isn't going to cause your Green Cheek any issues at all, though as Noodles mentioned you CANNOT put your Green Cheek's cage or any play-stands, floor/tabletop perches etc. near any AC units, Fans, air-vents that are in the apartment floors, walls, ceilings, etc...And make sure that his cage is not anywhere close enough to a window that he can feel the sunlight or the heat from the sunlight, because many, many pet birds have died of Heat-Exhaustion/Heat-Stroke due to their cages being close enough to windows that the heat from the sunlight coming-in has literally cooked them. It's fine to put their cages somewhere adjacent to a window if they like to look outside during the day while no one is home, but you just have to make sure that the rays from the sun cannot actually cause your bird or your bird's cage to feel the heat...
And just as an FYI that needs to be said, only because I've heard many Bird owners and Reptile and/or Amphibian owners mention that they do this during the summer months when the sunlight is bright and regularly shines through the windows of their homes...UVB light-rays CANNOT AND DO NOT PENETRATE THROUGH ANY TYPE OF GLASS OR CLEAR-PLASTIC!!! All types of glass, clear-plastic, plexi-glass, etc. BLOCK 100% OF ALL UVB LIGHT-RAYS FROM BOTH NATURAL SUNLIGHT AND FROM ARTIFICIAL UVB BULBS/LIGHT-TUBES THAT YOU BUY FOR YOUR PET REPTILES AND AMPHIBIANS! I've heard many, many bird/parrot owners make statements such as "I like to put my bird's cage in front of the windows in the house so that they can get the benefits of natural-sunlight without having to go outside"...And unfortunately most people aren't aware that 100% of the UVB light emitted by the Sun and any artificial UVB lights is blocked completely by the window-glass, so there is absolutely NO BENEFIT AT ALL to putting your bird, reptile, amphibian, or any other living creature in front of a window so they can "get some sunlight" (in-terms of the health-benefits of natural-sunlight or UVB light from an artificial UVB light), because the beneficial light-rays from the Sun are the UVB light-rays, and they cannot at all penetrate through any type of glass, plexi-glass, clear-plastic covers/lids, etc...In addition, UVB-LIGHT ALSO CANNOT PENETRATE THROUGH MESH OR SCREEN, SUCH AS THE TYPE THAT PEOPLE HAVE IN THEIR SCREEN-DOORS, OR THAT THEIR WINDOW-SCREENS ARE MADE OF!!! So opening up the window glass to allow the sunlight to come through the window-screens for your bird, reptile, etc. will also NOT provide them with any UVB-light unfortunately (UNLESS the holes in the mesh/screen are very large, which most window-screens, storm-door screens, screen-lids that go over tanks/vivariums, AS WELL AS MOST MESH/SCREEN THAT IS USED AROUND SCREENED-IN PORCHES AND SUN-PORCHES/SUN-ROOMS. Obviously the mesh/screening that is used in window-screens, door-screens, and around screened-in porches, patios, sun-rooms, etc.typically has extremely small holes in it because that's the only way to keep out the insects, spiders, and other critters, and having mesh/screening with large enough holes to allow UVB-light to pass through it would also allow all kinds of insects, spiders, etc. right into your home or your Screened-in Porches or Sun-Rooms...So the point is that it's not AT ALL worth risking heat-exhaustion/heat-stroke by putting your bird's, reptiles, etc. cages or stands close to windows or on screened-in porches because you want them to be able to absorb natural sunlight, because they aren't absorbing ANY BENEFICIAL SUNLIGHT AT ALL if they are behind glass, plexi-glass, clear-plastic, or mesh/screen.
(I think it was just last-year we had a member who lost both of his Lovebirds because he would keep their cage out on his balcony so they could "get natural sunlight", and his balcony was screened-in with the same fine-mesh that many, many balconies, decks, porches, etc. are screened-in with, which blocks 100% of the UVB-light from the sun anyway...And both of his Lovebirds died of heat-stroke, and for no reason at all...So this is just an FYI, if it can educate even 1 person and change what they've been doing with their birds and save just 1 bird life, then it's worth posting it)...
****To the OP...As Noodles already mentioned above/asked you, my main concern regarding your post is in-regard to what you said about "putting a light-sheet over your bird's cage", and "getting him used to being covered"...You are talking about covering your bird's cage with a "light-sheet" ONLY during nighttime while he's sleeping, correct? Because you should NEVER, EVER, EVER cover your bird's cage during the daytime with anything, just as an FYI. A lot of bird-owners mistakenly (and unknowingly/accidentally, just because they don't know any better) cover their bird's cages with something during the daytime-hours when they aren't at home, or at times they want them to be quiet and not bother them or their spouses (maybe because they work from home, etc.), or for whatever the reason, it doesn't matter, it's extremely cruel to a bird to keep them covered during the daytime-hours, and not only is it cruel and unfair to the bird, it's also extremely unhealthy for them physically, psychologically, and behaviorally in-regards to their sex-hormones...