Two deterministic finite automata are almost equivalent if they disagree in acceptanceonly for finitely many inputs. An automaton A is hyper-minimized if noautomaton with fewer states is almost equivalent to A. A regular languageL is canonical if the minimal automaton accepting L ishyper-minimized. The asymptotic state complexitys∗(L) of a regular languageL is the number of states of a hyper-minimized automaton for a languagefinitely different from L. In this paper we show that: (1) the class ofcanonical regular languages is not closed under: intersection, union, concatenation,Kleene closure, difference, symmetric difference, reversal, homomorphism, and inversehomomorphism; (2) for any regular languages L1 andL2 the asymptotic state complexity of their sumL1 ∪ L2, intersectionL1 ∩ L2, differenceL1 − L2, and symmetricdifference L1 ⊕ L2 can be boundedbys∗(L1)·s∗(L2).This bound is tight in binary case and in unary case can be met in infinitely many cases.(3) For any regular language L the asymptotic state complexity of itsreversal LR can be bounded by2s∗(L). This bound is tightin binary case. (4) The asymptotic state complexity of Kleene closure and concatenationcannot be bounded. Namely, for every k ≥ 3, there exist languagesK, L, and M such thats∗(K) = s∗(L) = s∗(M) = 1ands∗(K∗) = s∗(L·M) = k.These are answers to open problems formulated by Badr et al.[RAIRO-Theor. Inf. Appl. 43 (2009) 69–94].