How to make sense of this integral?

On page 10 of this paper on quantum field theory. There is the integral:

$$ln(N(T)) propto iiintlimits_{-infty}^{infty} ln (sinh(T omega_k)) d^3k$$

where $omega_k$ is the energy $sqrt{|k|^2+m^2}$

Now cleary this integral is divergent. Is there any way to make sense of this integral to give function $N(T)$ that is not infinite?

For example I notice that for large $T$ and keeping the limits large but not infinite , $ln(N(T))approx c T$ where $c$ is a very large number. (Perhaps dividing by a constant $c$ which depends on the limits of integration would give a finite function?)

Edit:

My guess would be that this is proportional to:

$$ln(N(T)) propto lim_{Lambdarightarrow infty} frac{
iiintlimits_{-Lambda}^{Lambda} ln (sinh(T omega_k)) d^3k
}{
iiintlimits_{-Lambda}^{Lambda} ln (sinh( omega_k)) d^3k
}$$

Which would give $ln(N(1))=1$ so that’s a finite value and hopefully this would give finite values for other values of $T$. But I can’t prove it.