What's the highest temperature that algae can survive at?
Based on the habitat of Algaes, a classification called "Thermophytes" can bear high temperature. The Thermophyte algaes use to grow around hot springs and can live around 70°C.

Please find below the reference for all classifications and more details.

http://www.yourarticlelibrary.com/biology/8-types-of-algae-classified-according-to-their-habitats/7098/

http://www.biologydiscussion.com/essay/essay-on-algae/20891