public class GameUI : MonoBehaviour
{
    public Text timeLabel;

    private bool isInitialized = false;

    void Start()
    {
        isInitialized = true;
        GameController.instance.OnGameInfoChanged += this.OnGameInfoChanged;
    }

    void OnEnable()
    {
        if (isInitialized)
            GameController.instance.OnGameInfoChanged += this.OnGameInfoChanged;
    }

    void OnDisable()
    {
        GameController.instance.OnGameInfoChanged -= this.OnGameInfoChanged;
    }

    void OnGameInfoChanged(GameType type)
    {
        //...
    }
}