Halt.
heartless
hate
Haste is a synonym for in a hurry. It begins with the letter h.
Something to do with help: helping, helpfulness
Heavenly is a synonym for tasty.
hot
hospital
http://thesaurus.reference.com/browse/boastful There are some good entries here, just look at all the h-words.
HOP is a verb right? It also starts with letter "H"
The phrase "to lift" means to pick something up. There are several different words that can be substituted for it. Synonyms for it that begins with the letter "H" would be heave, heft, hike, and hoist.
A physical change that starts with the letter H is humidify.
Ill-fortune, hardship, and pileup.