A vs. An with words that start with H.

In the past decade or so, I’ve been peeved by the way people think they should treat “an” preceding words that start with H. I’ve heard news anchors referring to “an hospital, etc.,” and it makes me cringe. Here’s the deal: If the word has the H sound as in hospital, use “a.” If the word drops the H sound as in honor, treat is as you would any word that starts with a vowel. Example: An honor. An apple.