Clearly, government agencies should default to observing the broader trends of API producers but are there particular best practices that a government API should follow?
(Disclaimer - I am the Sr. API Strategist for GSA)
You don't identify what kind of areas you're looking for advice on. But I'll highlight several that I think are particularly relevant for Government sources:
Where feasible, data should not just be made available via an API, but also available for download. This supports other kinds of uses. I think this is important as one goal of Open Government Data is to drive innovation and creation of new businesses. This is easier to do if the entire dataset is available. For example third-parties can offer value-added services over that data.
APIs should be accessible without usage restrictions or API keys. This avoids having any barriers to entry. Scaling may be an issue, but it'd be better to offer an open, unsupported API and then offer alternatives for higher-volume usage (and this is the kind of value-add that a third-party can offer)
Data must be clearly licensed. And that licensing information should be included not just in the API documentation but discoverable from the data itself. E.g. linked in API responses. Users should be crystal clear on how they can re-purpose the data. This is important for all Open Data, but particularly so for government.
Use of Open Standards. Again, this is a benefit in all cases, but government providers should avoid using proprietary formats or data protocols that might impede usage.
Versioning of your API's, please!