"building for worship, edifice dedicated to the service of a deity or deities," Old English tempel, from Latin templum "piece of ground consecrated for the taking of auspices, building for worship of a god," of uncertain signification.
Commonly referred to PIE root *tem- "to cut," on notion of "place reserved or cut out" [Watkins], or to root *temp- "to stretch" [Klein, de Vaan], on notion of "cleared (measured) space in front of an altar" (from PIE root *ten- "to stretch;" compare temple (n.2)), the notion being perhaps the "stretched" string that marks off the ground. Compare Greek temenos "sacred area around a temple," literally "place cut off," from stem of temnein "to cut." Figurative sense of "any place regarded as occupied by divine presence" was in Old English. Applied to Jewish synagogues from 1590s.
"flattened area on either side of the forehead," mid-14c., from Old French temple "side of the forehead" (11c.), from Vulgar Latin *tempula (plural taken as fem. singular), from Latin tempora, plural of tempus (genitive temporis) "side of the forehead," generally accepted as having originally meant "the thin stretch of skin at the side of the forehead" and being from PIE *temp- "to stretch," an extension of root *ten- "to stretch." The sense development would be from "stretchings" to "stretched skin."
A similar notion seems to be at work in Old English ðunwange, Old Norse þunn-vangi, Old High German dunwangi "temple," literally "thin cheek." The less-likely guess is that it is associated with tempus span "timely space" (for a mortal blow with a sword).