In iOS 7, some apps can now update their data even when the app is not open. This feature is called "Background App Refresh". How often does each app update their content using this method? Is it up to the app to determine the frequency, or does iOS 7 limit them?
This depends on several factors and is not generally tied to a fixed time schedule - active usage of the device, push notifications to apps, location based updates and iOS 7's learning ability to optimize based on your usage patterns for apps.
From iOS: Understanding multitasking (bold italic emphasis mine):
Certain tasks or services can continue to run in the background. To lessen the affect on battery life, normal app background refreshing is scheduled for efficient times, such as when your device is connected to Wi-Fi, plugged into a power source, or being actively used.
You can adjust your apps' background activity in Settings > General > Background App Refresh.
When Background App Refresh is on, apps that take advantage of this feature can refresh themselves in the background. For example, an app can check if new content is available and download the updates, or retrieve the updated content in the background when it receives a push notification, so the new content is ready for viewing when you launch the app. Apps can also schedule background refreshing based on your location. iOS learns patterns based on your use of the device and tries to predict when an app should be updated in the background. It also learns when the device is typically inactive, such as during the night, to reduce update frequency when the device not used.
To improve battery life, you could selectively turn off background refresh for certain apps from Settings > General > Background App Refresh.