Why does the type-A subdivision algebra look like the Rota-Baxter algebra axiom? Let $\mathbf{k}$ be a commutative ring, and $\beta$ an element of $\mathbf{k}$. Fix a positive integer $n$, and set $\left[n\right] = \left\{1,2,\ldots,n\right\}$.
The $n$-th type-A subdivision algebra over
$\mathbf{k}$ for parameter $\beta$ is the commutative $\mathbf{k}$-algebra
$B$ with


*

*generators $x_{i,j}$ indexed by all the $n\left(  n-1\right)  /2$ pairs
$\left(  i,j\right)  $ of integers satisfying $1\leq i<j\leq n$;

*relations $x_{i,j}x_{j,k}=x_{i,k}\left(  x_{i,j}+x_{j,k}+\beta\right)  $ for all $\left(  i,j,k\right)  \in\left[  n\right]  ^{3}$ satisfying $i<j<k$.
Alternatively, we can define $B$ in a more symmetric fashion: Namely, $B$ is
the commutative $\mathbf{k}$-algebra with


*

*generators $x_{i,j}$ indexed by all the $n\left(  n-1\right)  $ pairs
$\left(  i,j\right)  $ of distinct integers in $\left[n\right]$;

*relations $x_{i,j}+x_{j,i}=-\beta$ whenever $i\neq j$, as well as
$x_{i,j}x_{j,k}+x_{j,k}x_{k,i}+x_{k,i}x_{i,j}+\beta\left(  x_{i,j}
+x_{j,k}+x_{k,i}\right)  +\beta^{2}=0$ whenever $i,j,k$ are distinct elements
of $\left[n\right]$.
The $\mathbf{k}$-algebra $B$ has appeared in various contexts. It was
originally introduced by Karola Mészáros as the abelianization of
Anatol Kirillov's quasi-classical Yang-Baxter
algebra.
It is a deformation of the Orlik-Terao algebra of the braid arrangement of
type $A_{n-1}$ (with the case $\beta=0$ corresponding to the Orlik-Terao
algebra). It is probably isomorphic to a $\mathbf{k}$-subalgebra of the
localization of the polynomial ring $\mathbf{k}\left[  q_{1},q_{2}
,\ldots,q_{n}\right]  $ at the multiplicative subset generated by the
differences $q_{i}-q_{j}$ for $i<j$ (here I say "probably" because I can only
show this for $\beta=0$, in which case it is isomorphic to the $\mathbf{k}
$-subalgebra generated by all $\dfrac{1}{q_{i}-q_{j}}$). It comes up in the
computation of volumes of flow polytopes
and evaluations of Grothendieck
polynomials. See my recent preprint
arXiv:1704.00839
for more on it.
On the other hand, recall that a Rota-Baxter algebra of weight $\beta$ means a $\mathbf{k}$-algebra $R$ equipped with a $\mathbf{k}$-linear map $P:R\rightarrow R$
(called its Rota-Baxter operator) that satisfies
\begin{equation}
P\left(  a\right)  P\left(  b\right)  =P\left(  P\left(  a\right)  b\right)
+P\left(  aP\left(  b\right)  \right)  +\beta P\left(  ab\right)
\label{eq.rota-baxter.def} \tag{1}
\end{equation}
for all $a,b\in R$. (Some authors, like those of the Wikipedia
page, prefer to put the
$\beta P\left(  ab\right)  $ addend on the left instead of the right hand
side, but this just boils down to replacing $\beta$ by $-\beta$.)
The axiom \eqref{eq.rota-baxter.def} of the Rota-Baxter algebra is uncannily
similar to the relations
\begin{equation}
x_{i,j}x_{j,k}=x_{i,k}\left(  x_{i,j}+x_{j,k}+\beta\right)  \label{eq.relB} \tag{2}
\end{equation}
of the algebra $B$. Indeed, represent each monomial in the indeterminates
$x_{i,j}$ as a multigraph on the vertex set $\left\{  1,2,\ldots,n\right\}  $,
where each indeterminate $x_{i,j}$ appearing in the monomial contributes an
edge $ij$ to the multigraph. Then, \eqref{eq.relB} can be visually rewritten
as

(where all vertices other than $i,j,k$ are omitted). Now, imagine writing an
"$a$" between the $i$ and the $j$, and writing a "$b$" between the $j$ and the
$k$, and interpreting each edge as a signal to apply $P$ to whatever stands
under the edge. The above equality thus becomes

which is precisely \eqref{eq.rota-baxter.def}.

Question. Can this resemblance be turned into anything concrete (e.g., an
  action of $B$ on Rota-Baxter algebras?). Barring that, can we define "Rota-Baxter algebras of other types"?

 A: This is too long for a comment.
The reason for my question in comments was as follows. In some cases (Gerstenhaber algebras, Poisson algebras, etc.) an operad describing a certain algebraic structure is an operad in the category of cocommutative coalgebras (some people refer to that as a Hopf operad). Alternatively, one can think about a collection of commutative associative algebras that form a cooperad. For your algebras, the most obvious candidate for cooperad structure does lead to an honest cooperad structure only for $x_{ij}x_{ji}=0$. But in this case this actually leads to an operad of associative algebras presented via unusual generators, see https://arxiv.org/abs/math/0412206 for instance. 
A Hopf operad structure on an operad is the same as a functorial rule for defining tensor products for algebras over that operad. I think some people study tensor products of Rota-Baxter algebras, but maybe in some way that their tensor products are not RB structures on the tensor products of the underlying vector spaces. However, if you know a way to equip the tensor product of RB algebras with a RB structure, this would give you a natural sequence of (co)algebras arising from it. 
