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The KT transition has a special properties that, during the phase transition the heat capacity stay finite (so the behaviour of the heat capacity cannot reflect any critical behaviours). However, the correlation length and the susceptibility diverge. Specifically, $\xi \sim exp(b |t|^{-\nu})$ and $\chi \sim \xi^{2-\eta}$, where $t$ is the reduced temperature, $b$ is some constant, and $\nu$ and $\eta$ are critical exponents.

I understand that, in this case the correlation diverges so fast comparing to the first/second order phase transition, but how this leads to the non-divergence behaviour of the heat capacity?

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I am not too familiar with KT transitions yet, but I would like to learn about them myself.

I have read in the notes of Prof. Jensen (available online http://www.mit.edu/~levitov/8.334/notes/XYnotes1.pdf) in the end of chapter 4.2 that the divergence in the specific heat is so fast that it is experimentally not observable.
Analytically (according to his formulae) there would be a divergence in the specific heat c ~ exp(a/sqrt(T-Tc)) where a is a constant (depending on the dimension of the system). I think normally in phase transitions the divergence is a polynomial and not exponential in (T-Tc)- that's probably why he claims that the divergence is so fast.

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