Yes, the Nest can do this. They offer an iOS and Android app to control it via a phone/tablet, and it can also be controlled by logging in at nest.com. When setting up the thermostat, it will ask to join a WiFi network, and then help you bind it to a Nest.com account.

If the thermostat is mounted in a spot in the apartment where it's motion sensors can detect people there, it will also enable auto away mode. If they forget to turn it down one visit, it will do so on it's own if they want.