One reason that Lawvere theories might be useful is in homotopy theory: Badzioch has apparently done work on formalizing the notion of a "homotopy algebra" (over a given Lawvere theory). The idea seems to be that an algebra over a Lawvere theory is something you can make homotopyish without recourse to things like operads: given $T$, a homotopy $T$-algebra is a functor $T \to \mathrm{Spaces}$ which preserves products up to homotopy equivalence (rather than on the nose).

In general, it seems that having a categorical language rather than explicit operations is much better for making algebraic structures homotopy invariant. An earlier example is given by the $\Gamma$-spaces of Segal: one writes the axioms of an abelian monoid in terms of a suitable product-preserving functor $\mathrm{Fin}_* \to \mathbf{Sets}$ and replaces those then by weakly product-preserving functors to spaces. It turns out grouplike $\Gamma$-spaces are essentially equivalent to infinite loop spaces.

I don't understand too much of this yet, but what's a little puzzling to me is that homotopy $T$-algebras (according to Badzioch) turn out to be essentially the same as ordinary $T$-algebras. (By contrast, the homotopy-theoretic analog of an abelian group--an infinite loop space--is generally very different from an abelian group.)

One reason that Lawvere theories might be useful is in homotopy theory: Badzioch has apparently done work on formalizing the notion of a "homotopy algebra" (over a given Lawvere theory):

• Algebraic theories in homotopy theory, Annals of Mathematics 155 Issue 3 (2002) 895-913, https://doi.org/10.2307/3062135

The idea seems to be that an algebra over a Lawvere theory is something you can make homotopyish without recourse to things like operads: given $T$, a homotopy $T$-algebra is a functor $T \to \mathrm{Spaces}$ which preserves products up to homotopy equivalence (rather than on the nose).

In general, it seems that having a categorical language rather than explicit operations is much better for making algebraic structures homotopy invariant. An earlier example is given by the $\Gamma$-spaces of Segal: one writes the axioms of an abelian monoid in terms of a suitable product-preserving functor $\mathrm{Fin}_* \to \mathbf{Sets}$ and replaces those then by weakly product-preserving functors to spaces. It turns out grouplike $\Gamma$-spaces are essentially equivalent to infinite loop spaces.

I don't understand too much of this yet, but what's a little puzzling to me is that homotopy $T$-algebras (according to Badzioch) turn out to be essentially the same as ordinary $T$-algebras. (By contrast, the homotopy-theoretic analog of an abelian group—an infinite loop space—is generally very different from an abelian group.)