Area Republican periodically studies cool style (3,4)
I believe the answer is:
art deco
'style' is the definition.
(decorative style)
'area republican periodically studies cool' is the wordplay.
'area' becomes 'a' (maths abbreviation).
'republican' becomes 'R' (abbreviation in US politics).
'periodically' means one should take alternating letters (letters must be selected 'every so often').
'studies'+'cool'='studiescool'
The alternate letters of 'studiescool' are 'tdeco'.
'a'+'r'+'tdeco'='ART DECO'
(Other definitions for art deco that I've seen before include "Nineteen twenty's and thirties style" , "nineteen thirty's style" , "Style characterised by bold geometric shapes" , "Highly geometric style" , "Redcoat (anag) -- inter-war design style" .)