getSupportActionBar with YouTubeBaseActivity

YouTubeBaseActivity extends Activity, (as opposed to, for example AppCompatActivity), so the getSupportActionBar() method doesn't exist.

You could try to make your class extend AppCompatActivity, and use a YouTubePlayerSupportFragment instead wherever you would normally use a YouTubePlayerView.

Edit:

Add the following to your layout file, in place of a YouTubePlayerView

<fragment
    android:name="com.google.android.youtube.player.YouTubePlayerSupportFragment"
    android:id="@+id/youtube_fragment"
    android:layout_width="match_parent"
    android:layout_height="wrap_content"/>

Access it in onCreate() of your Activity in the same way you would any other static Fragment

public class CustomYouTubeActivity extends AppCompatActivity implements YouTubePlayer.OnInitialisedListener {

    @Override
    public void onCreate(Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);

        setContentView(R.layout.fragments_demo);

        YouTubePlayerSupportFragment frag =
            (YouTubePlayerSupportFragment) getSupportFragmentManager().findFragmentById(R.id.youtube_fragment);
        frag.initialize(DeveloperKey.DEVELOPER_KEY, this);
    }

    @Override
    public void onInitializationSuccess(YouTubePlayer.Provider provider, YouTubePlayer player,
  boolean wasRestored) {
        if (!wasRestored) {
        //I assume the below String value is your video id
        player.cueVideo("nCgQDjiotG0");
    }

    @Override
    public void onInitializationFailure (YouTubePlayer.Provider provider, YouTubeInitializationResult error) {
        if (errorReason.isUserRecoverableError()) {
            errorReason.getErrorDialog(this, RECOVERY_DIALOG_REQUEST).show();
        } else {
            String errorMessage = String.format(getString(R.string.error_player), errorReason.toString());
            Toast.makeText(this, errorMessage, Toast.LENGTH_LONG).show();
        }
    }    

}