Motivated by fractal geometry of self-affine carpets and sponges,
Feng--Huang (2016) introduced the theory of weighted topological entropy
for factor maps between dynamical systems.
We introduce a new approach to this theory.
Our new definition of the weighted topological entropy is very different from
the original definition of Feng--Huang.
The equivalence of the two definitions seems highly nontrivial.
Their equivalence can be seen as a generalization of the dimension formula for the Bedford--McMullen carpets
in purely topological terms.