I think the title says it all but if needed - I'll edit this. Of course I googled but I got nowhere. Google results were total shit. Google won't help me if I don't know the exact name for that fucking thing. It's a very stupid design choice. Five regular old push buttons would be so much better.
About an hour ago I managed to stick it back on but I'm nearly 100% sure it's going to fall off again in week or two if not far sooner.
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Edit
I don't mean part of the unit itself. I mean just the thin plastic cover over the buttons that's ~3mm thick.
That's a Honeywell T6. It's really easy to replace, theres a separate part that retains the wires that you can keep attached to the wall. IIRC, it's just one screw on the bottom and then it pops off.
You can find open box/used ones on eBay for less than $50. There's an installer menu on these thermostats, go into that and mark down the settings, and set the new one up the same way. You'll find a PDF of the installers manual online that can tell you how to get into the menu, or I can help you.
You can probably glue it back on with some super glue. Just a little on the edges of the rubbery cover should do it.
If you want something less permanent just slap some clear packing tape over it. You'll still have enough "give" in the tape to push the buttons and since it's clear you can read what's under the tape. Then if you want to try a bigger fix later you can remove the tape.
I would call it either a control panel or a button panel, but would defer to whatever the owner's manual calls it in a schematic.
bizarrely, those thermostat controllers can be pretty cheap to replace and totally in the realm of DIY.
the wires leading to HVAC should be color coded and sometimes are tagged.
just saying if trying to fix a cheap component becomes too infuriating (because they are often cheaply made).
anyway, good luck
It's just buttons, in this case in a kind of soft silicone formed into one piece I'm guessing with extrusions on the other side to depress contacts? You can find similar things in some remotes and other devices. I would try and use superglue or an epoxy suitable for bonding plastic to other things and try drawing a thin line around the outside of it in a rectangle towards the edges. If you want to be extra careful you might try just gluing the thin sides and bottom of it to hold it in place without gluing the back.
As someone else said you can also just use clear tape, make sure it has a tiny bit of give but not enough to cause ridges or bubbles and you should have no issue using it behind something like clear packing tape.
Id just replace the whole unit. Turn off the breaker power, use a touchless tester to verify it’s off, take pictures of the wires and where they’re connected, disconnect.
Edit: I’ve read Ecobee thermostats are good
My plan is to use a cotton swab to very gingerly push the buttons. I assume the panel is less likely to fall off. Why? I only did a few test pushes with a swab but it seems better than a fingertip. The swab focuses the push right on the button and the swab seemed not to be nearly as sticky to the panel as my fingertip.
Why didn't they make it just five fucking buttons? Sigh.
Is the entire panel falling off or is it just a cover for the buttons? If its the cover then maybe you can stick some sort of plastic caps on the panel to replace the gray cover. If its the entire panel then go with came apart at Kmart's suggestion.