DIAGNOSIS Yeah, this one's common. It's usually a weak or unstable Wi-Fi signal at the thermostat's location, or a router setting that the thermostat doesn't play nice with.
FIX Start with the simple stuff. On the thermostat, go to Menu > Wi-Fi > Network Status. It'll show the signal strength. If it's one or two bars, that's your problem. You can try moving your router, using a Wi-Fi extender, or if possible, connect your thermostat to your router's 2.4 GHz network (not the 5 GHz one). The RTH9585WF only works on 2.4 GHz. Also, in your router settings, make sure you don't have 'AP Isolation' or 'Client Isolation' turned on, as that blocks device-to-device communication the app needs.
IF THAT DIDN'T WORK The thermostat's internal network connection can get hung up. Do a full network reset on the device. From the thermostat, go to Menu > Wi-Fi > Clear Network. Then set it up again from scratch. This clears any corrupted connection data.
HEADS UP If your router automatically swaps between 2.4GHz and 5GHz with the same network name (SSID), it can confuse the thermostat. Create a separate, dedicated 2.4GHz network with a different name (like 'HomeWiFi_2G') and connect the thermostat to that. It's more stable.