If "one", "two", and "three" are cardinal numbers, and "first", "second", and "third" are ordinal numbers, then what are "once", "twice", and "thrice"? Is there a name for this kind of number?
One longtime term for this group of words is numeral adverb. William Fowler, English Grammar: The English Language in Its Elements and Forms (1850) uses this nomenclature:
§ 193. NUMERALS express the relation of Number and Quantity. They are divided into the following classes:
IV. NUMERAL ADVERBS, which answer the question How often? as, Once, twice, thrice, four times.
That terminology goes back to at least 1822, when it was used by Edward Everett, in his translation of Philipp Buttmann, Greek Grammar.
The same terminology is used as recently as 2007, by Alexander Coupe, A Grammar of Mongsen Ao (2007):
Numeral adverbs are derived by attaching the suffix –pən to the stem of a cardinal number. The numeral adverb khən ‘once’ is irregular, in common with both the distributive and ordinal derivations of ONE, and does not use this suffix to express its adverbial meaning. Apart from [a complication involving the word for ‘twice’], the derivations of the remaining numeral adverbs are completely regular, e.g. à-sə̀m-pən 'thrice', phə̀lì-pən 'four times', ...