Connection between algebraic logic and computational complexity of logics? I'm learning a bit about algebraic logic and I was wondering how knowing the algebraic semantics of a given logic might help the study of the logic itself from a computational point of view.

In particular, is there any example of a complexity (or decidability) result for the satisfiability problem for some logics that was obtained by reasoning about its algebraic semantics?

For example, the semantics of propositional logic can be given in terms of boolean algebras. Is there any connection between them and the fact that SAT is decidable and $NP$-complete?
The example you gave extends as follows:


SAT for arbitrary lattices (meaning, is a given formula satisfiable in some lattice) is polynomial-time decidable
SAT for modular lattices is Turing undecidable:
Freese, Ralph, Free modular lattices, Trans. Am. Math. Soc. 261, 81-91 (1980). ZBL0437.06006.
SAT for Boolean algebras is NP-complete


So while modular lattices are in between Boolean algebras and arbitrary lattices, they are the most complicated.