Suppose consumption demand of GDP decreases during a recession while the other demands are unaffected. What happens to the equilibrium real interest rate in the economy?

Since consumption represents almost 70% of the GDP, any change in consumption affects the economy more than any change in the rest of the components of the GDP (net exports, investment, government). If consumption decreases, then the real interest rate will decrease.

The higher the interest rate, the lower the consumption level. This should increase the savings = more investment in the economy, but since consumption is so important to the economy, a decrease in consumption will decrease the equilibrium interest rate. This lowering in the real interest rate will be carried out in order to try to increase consumption.

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