Infinite sum involving $\frac{\binom{2n}{n}}{2^{2n}}$ resulting in gamma function product

Question

I enjoy sums featuring $$\rm{T}(n) = \frac{\binom{2n}{n}}{2^{2n}} = (-1)^n \binom{1/2}{n},$$ and I'm on the lookout for techniques of handling sums of these quantities.

For a couple examples, MSE 2797610 and MSE 2797729 have helped show me a couple ways to handle such sums, such as manipulating generating functions.

My question here is: How can I show that $$\sum_{n=0}^{\infty} \rm{T}(2n)\rm{T}(n)^2 = \frac{\pi}{\big(\Gamma(\frac{5}{8})\Gamma(\frac{7}{8})\big)^2 } ?$$

Such a sum arose when I attempted to evaluate the integral $$\int_0^{2\pi} \int_0^{2\pi} \frac{dx dy}{\sqrt{2-\cos(x)-\cos(y)}},$$ and was evaluated using Mathematica.

I think it has something to do with the hypergeometric function ${}_2F_1$ — Maximilian Janisch, Dec 14 '19 at 23:34
Observation: $$\sum_{n=0}^\infty T(2n) T(n)^2 = {}_3F_2\left(\frac14,\frac24,\frac34;1,1;1\right),$$ where $${}_3F_2$$ is the generalized hypergeometric function — Maximilian Janisch, Dec 15 '19 at 00:03
I am pretty sure that there is some known identity which allows you to reduce this to $$\frac{\pi}{\big(\Gamma(\frac{5}{8})\Gamma(\frac{7}{8})\big)^2 } $$ — Maximilian Janisch, Dec 15 '19 at 00:04
For example see here: http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric3F2/03/02/01/ — Maximilian Janisch, Dec 15 '19 at 00:07
@MaximilianJanisch Good thinking. If you write up your comments as an answer, I'll accept it. I don't have much background with hypergeometric identities, so I'd appreciate a link with a little more discussion in your answer. — user196574, Dec 15 '19 at 09:09

Infinite sum involving $\frac{\binom{2n}{n}}{2^{2n}}$ resulting in gamma function product

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