calculus - Multiple integrals involving product of gamma functions

The following integral was posted a few days back on Integrals and Series forum:

$$\int_0^{2\pi} \int_0^{2\pi} \int_0^{2\pi} \frac{dk_1\,dk_2\,dk_3}{1-\frac{1}{3}\left(\cos k_1+\cos k_2+ \cos k_3\right)}=\frac{\sqrt{6}}{4}\Gamma\left(\frac{1}{24}\right)\Gamma\left(\frac{5}{24}\right)\Gamma\left(\frac{7}{24}\right)\Gamma\left(\frac{11}{24}\right)$$

I am curious if there is a closed form solution for:

$$\int_{\large[0,2\pi]^n} \frac{dk_1\,dk_2\,dk_3\,\cdots \,dk_n}{1-\frac{1}{n}\left(\cos k_1+\cos k_2+\cos k_3+\cdots +\cos k_n\right)}$$

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Since $\left|\dfrac{\cos k_1 + \cos k_2 + \cos k_3}{3}\right|<1$,

$$\int_0^{2\pi} \int_0^{2\pi} \int_0^{2\pi} \frac{dk_1\,dk_2\,dk_3}{1 - \frac 1 3 \left( \cos k_1 + \cos k_2 + \cos k_3 \right)}$$
$$=8\int_0^{\pi} \int_0^{\pi} \int_0^{\pi} \frac{dk_1\,dk_2\,dk_3}{1 - \frac 1 3 \left( \cos k_1 + \cos k_2 + \cos k_3 \right)}$$
$$=8\sum_{n=0}^{\infty} \frac{1}{3^n} \int_0^{\pi} \int_0^{\pi} \int_0^{\pi} \left( \cos k_1 + \cos k_2 + \cos k_3 \right)^n\,dk_1\,dk_2\,dk_3 $$

We can ignore the odd values of $n$ as the integral is zero for them. Also, for even values of $n$, the exponents of cosines in the expansion of $\left( \cos k_1 + \cos k_2 + \cos k_3 \right)^{2n}$ must be even. Hence, from multinomial therem, we can write:

$$8\sum_{n=0}^{\infty}\,\,\sum_{m_1+m_2+m_3=n} \frac{1}{3^{2n}}\frac{(2n)!}{(2m_1)! (2m_2)! (2m_3)!} \int_0^{\pi} \int_0^{\pi} \int_0^{\pi} \cos^{2m_1}k_1\cos^{2m_2}k_2 \cos^{2m_3}k_3\,dk_1\,dk_2\,dk_3$$

$$ = 16\sum_{n=0}^{\infty}\,\,\sum_{m_1+m_2+m_3=n} \frac{1}{3^{2n}}\frac{(2n)!}{(2m_1)! (2m_2)! (2m_3)!} \int_0^{\pi/2} \int_0^{\pi/2} \int_0^{\pi/2} \cos^{2m_1}k_1\cos^{2m_2}k_2 \cos^{2m_3}k_3\,dk_1\,dk_2\,dk_3$$

Using the result: $\int_0^{\pi/2} \cos^{2k}x\,dx=\frac{(2k)!}{4^k (k!)^2}\frac{\pi}{2}$, the integral is,

$$2\pi^3 \sum_{n=0}^{\infty}\,\,\sum_{m_1+m_2+m_3=n} \frac{1}{36^n}\frac{(2n)!}{(m_1!)^2 (m_2!)^2 (m_3!)^2}$$

I am stuck here.

Any help is appreciated. Thanks!