Thermodynamics of $f(R)$ Theories

This series of three lectures was presented at ``Escola de Cosmologia e Gravita\c{c}\~ao" \url{https://cosmosecontexto.org.br/ecg-inscricoes} and webcast at \url{https://shorturl.at/2ZSI7} -- in Portuguese, but slides in English. We will go through a brief review on $f(R)$ theories (in the metric approach) and the usual requirements for successful modifications of General Relativity. Then we will open a large parenthesis to talk about a non-standard approach to Phase Transitions: the Catastrophe Theory. Finally, we will connect the previous lectures to introduce a new Thermodynamic interpretation of $f(R)$ theories.