Why are real interest rates so non-volatile compared to investment? Intuitively I would guess that real interest rates are highly related to investment.

But there is a very low interest rate volatility, compared to investment:

log investment is about 20 times more volatile than real interest rates. The standard deviation of interest rates is about 0.15 of that of log gdp, while investment is about 3 times as much.

What explains this massive difference?
In most countries, interest rates are actively managed by the government or central bank in pursuit of various policy objectives (some closely linked to investment, some, like stable foreign exchange markets, not).  Investment in turn is determined by a wide range of factors other than interest rates.  Basically I don't think that your assumption of a close link is valid.