Can anything be said for the topology of a topological monoid?

A topological group is one in which the group operations (the multiplication and inverse) are continuous, or equivalently as a group object in $\mathbf{Top}$.

They are uniformisable and hence are completely regular & $R_0$.

Can anything similar i.e. with respect to the separation axioms be said about topological monoids?

The space $X=(\Bbb N,+,0)$ with the topology generated by the sets $$A_n=\{0,1,...,n\}$$ is a topological monoid. If $m,n$ are naturals, then the smallest neighborhood of $m+n$ is $A_{m+n}$, and it contains the sum $A_m+A_n$, so addition is continuous.
$X$ is $T_0$, but not $T_1$, so it isn't uniformizable.