@Sardorek, ran into the same problem, spent more than an hour, and finally figured it out.
The solution is to add the Expo SDK version in the app.json file. My version is 40.0.1. You can see it in package.json.
"react-native": "https://github.com/expo/react-native/archive/sdk-40.0.1.tar.gz"
Upgrading to version 40 has removed it from the app.json file.
Now, add the major.minor.0 version in the app.json file like:
"sdkVersion": "40.0.0"
Looks like you cannot include the maintenance version number. If you do, you will get this error.
Error: Invalid sdkVersion. Valid options are 7.0.0, 8.0.0, 9.0.0, 10.0.0, 11.0.0, 12.0.0, 13.0.0, 14.0.0, 15.0.0, 16.0.0, 17.0.0, 18.0.0, 19.0.0, 20.0.0, 21.0.0, 22.0.0, 23.0.0, 24.0.0, 25.0.0, 26.0.0, 27.0.0, 28.0.0, 29.0.0, 30.0.0, 31.0.0, 32.0.0, 33.0.0, 34.0.0, 35.0.0, 36.0.0, 37.0.0, 38.0.0, 39.0.0, 40.0.0, 41.0.0
Couldn't publish because errors were found. (See logs above.) Please fix the errors and try again.
Now, run expo start, and voila it works! Hope this helps.