hold the line vs keep up

hold the line and keep up both are verbs.

Word	Noun	Adjective	Verb	Adverb
hold the line	No	No	Yes	No
keep up	No	No	Yes	No

As verbs, keep up is a hypernym of hold the line; that is, keep up is a word with a broader meaning than hold the line:

hold the line: hold the line on prices; keep the price of something constant
keep up: keep in safety and protect from harm, decay, loss, or destruction

Other hypernyms of hold the line include conserve, maintain, preserve.

hold the line (verb)	keep up (verb)
hold the phone line open	prevent from going to bed at night
hold the line on prices; keep the price of something constant	keep informed
	maintain a required pace or level
	keep in safety and protect from harm, decay, loss, or destruction
	lengthen or extend in duration or space