  Let A be a finitely presented associative monomial algebra. We study the
category qgr(A) which is a quotient of the category of graded finitely
presented A-modules by the finite-dimensional ones. As this category plays a
role of the category of coherent sheaves on the corresponding noncommutative
variety, we consider its bounded derived category. We calculate the categorical
entropy of the Serre twist functor this derived category and show that it is
equal to logarithm of the entropy of the algebra A itself. Moreover, we relate
these two kinds of entropy with the topological entropy of the Ufnarovski graph
of A and the entropy of the path algebra of the graph. If A is a path algebra
of some quiver, the categorical entropy is equal to the logarithm of the
spectral radius of the quiver's adjacency matrix.
