Electrical code and/or common sense requirements for thermostat wire
The Code rules are in Parts I and III of Article 725

Thermostat wiring is the most commonly encountered example of what the NEC terms a Class 2 control circuit, and the rules for those are set out in Parts I and III of NEC article 725.  In your case, there are a few things to consider:


725.25 requires removal of accessible portions of abandoned control circuit cabling
Your work still needs to be "neat and workmanlike" as per 725.24 -- slobbering cable everywhere is no good
Strapping Class 2 cables to raceways for support is NFG -- 725.143 prohibits it
You don't need to support Class 2 cable in a wall cavity, however -- this is implied by 725.24 not mentioning any support requirements for concealed Class 2 cables