Do the words contemporary and contemplate relate to each other in any way?
The OED explains that contemplate comes from
L. contemplāre, orig. deponent contemplārī, to survey, observe, behold, consider, contemplate, f. con- + templum ‘an open place for observation, marked out by the augur with his staff’
And that contemporary comes from:
L. type contemporāri-us, f. con- together + tempus, tempor- time, temporārius of or belonging to time; the actual formations in L. were contemporālis and contemporāneus
So there seems no surface relationship between tempus and templum.