cookie.java

public class WebViewActivity extends Activity {

  WebView myWebView;

	private final static String TAG = "WebView";

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

		setContentView(R.layout.webview);

		myWebView = (WebView) findViewById(R.id.webview);
           
                new WebViewTask().execute();
        }

        private class WebViewTask extends AsyncTask<Void, Void, Boolean> {
		String sessionCookie;
		CookieManager cookieManager;

	        @Override
		protected void onPreExecute() {
			CookieSyncManager.createInstance(WebViewActivity.this);
			cookieManager = CookieManager.getInstance();
                        
			sessionCookie = new PersistentConfig(getApplicationContext()).getCookieString();
			if (sessionCookie != null) {
                                /* delete old cookies */
				cookieManager.removeSessionCookie(); 
			}
			super.onPreExecute();
		}
		protected Boolean doInBackground(Void... param) {
                        /* this is very important - THIS IS THE HACK */
			SystemClock.sleep(1000);
			return false;
		}
		@Override
		protected void onPostExecute(Boolean result) {
			if (sessionCookie != null) {
				cookieManager.setCookie("yourdomain.com", sessionCookie);
				CookieSyncManager.getInstance().sync();
			}
			WebSettings webSettings = myWebView.getSettings();
			webSettings.setJavaScriptEnabled(true);
			webSettings.setBuiltInZoomControls(true);
			myWebView.setWebViewClient(new WebViewClient() {
				@Override
				public boolean shouldOverrideUrlLoading(WebView view, String url) {
					return super.shouldOverrideUrlLoading(view, url);
				}
				
			});
			myWebView.loadUrl("https://yourdomain.com/anything");
		}
	}
}

does it still work with android 4.4 built API 21

The CookieSyncManager is deprecated from API21