  In this paper, we extend the notion of temporo-spatial differentiation
problems to the setting of actions of more general topological groups. The
problem can be expressed as follows: Given an action $T$ of an amenable
discrete group $G$ on a probability space $(X, \mu)$ by automorphisms, let
$(F_k)_{k = 1}^\infty$ be a F{\o}lner sequence for $G$, and let $(C_k)_{k =
1}^\infty$ be a sequence of measurable subsets of $X$ with positive probability
$\mu(C_k)$. What is the limiting behavior of the sequence
  $$\left( \frac{1}{\mu(C_k)} \int_{C_k} \frac{1}{|F_k|} \sum_{g \in F_k} f(T_g
x) \mathrm{d} \mu(x) \right)_{k = 1}^\infty$$
  for $f \in L^\infty(X, \mu)$? We provide some positive convergence results
for temporo-spatial differentiations with respect to ergodic averages over
F{\o}lner sequences, as well as with respect to ergodic averages over
subsequences of the integers (e.g. polynomials), multiple ergodic averages, and
weighted ergodic averages.
