Trick for calculating the thermal entropy of the SYK model in the large $N$ limit

Reading about the SYK model I encounter a trick that should help to calculate the thermal entropy of the system. I am not able to understand what they are doing though. Particularly the part from eq.135 to eq.137 of Gabor Sarosi is giving me problems.
My attempt to understand this part:

$$e^{-beta F}=Zsim e^{-NI[G_*,Sigma_*]}$$

Where

$I[G,Sigma]=-frac{1}{2}logdet[partial_{tau}-Sigma]+frac{1}{2}intint dtau dtau’left(Sigma(tau,tau’) G(tau,tau’)-frac{1}{q}J^2G(tau,tau’)^qright)$

Since we don’t want to deal with the determinant we write

begin{equation}
begin{split}
Jpartial_Jleft(-frac{beta F}{N}right)&=Jpartial_J(-I)\
&=frac{J^2}{q}intint_0^beta dtau’dtau G_*(tau,tau’)^q\
&=frac{beta J^2}{q}int_0^beta dtau G_*(tau)^q
end{split}
end{equation}

Where in the last step I used translational invariance $G(tau,tau’)=G(tau-tau’)$. I use the Swinger-Dyson equations

begin{split}
&G(iomega_n)=[-iomega_n-Sigma(iomega_n)]^{-1}\
&Sigma(tau)=J^2G(tau)^{q-1}.
end{split}

from this one gets that

$G(tau)^q=frac{1}{J^2}Sigma(tau)G(tau)$

Plugging this in to our integral gets us:

begin{split}
Jpartial_Jleft(-frac{beta F}{N}right)&=frac{beta}{q}int_0^beta dtau Sigma(tau)G(tau)\
&=frac{1}{q}sum_{omega_n,omega_m}Sigma(iomega_n)G(iomega_m)frac{1}{beta}int_0^beta dtau e^{-itau(omega_n+omega_m)}\
&=frac{1}{q}sum_{omega_n}Sigma(iomega_n)G(-iomega_n)\
&=-frac{1}{q}sum_{omega_n}Sigma(iomega_n)G(iomega_n)\
&=frac{1}{q}sum_{omega_n}(iomega_n+G(iomega_n)^{-1})G(iomega_n)\
&=frac{1}{q}sum_{omega_n}iomega_n G(iomega_n)+frac{1}{q}sum_{omega_n}1\
&=-frac{beta}{q}lim_{taurightarrow0^+}partial_tau G(tau)+infty
end{split}

This is the point where I run in to trouble. Gabor Sarosi states that

$frac{J^2beta}{q}int_0^beta dtau G(tau)^q=-frac{beta}{q}lim_{taurightarrow0^+}partial_tau G(tau)$

Why doesn’t he have this infinite constant? Note that one can proof that $$frac{beta}{q}lim_{taurightarrow0^+}partial_tau G(tau)=frac{beta}{N}langle H rangle$$
Is he absorbing the infinite constant into the energy somehow? Maybe using some sort of regularization scheme I am unaware of? Or did I just make a mistake? Hopefully one of you guy’s can tell me what’s going on here.