Rationale and Methodology for Discounting the Investment in an NPV Calculation It is clear to me why the benefits accruing from an investment should be discounted. What I wish to understand is what the correct way discount the investment itself is.
Assuming zero inflation, consider that at a personal discount rate of $\rho$, the net present value of an investment $K$ can be computed as
$$
 \frac{R}{\rho - \mu} - K\ ,
$$
where $R$ is the expected revenue flow with trend $\mu$. If this quantity is positive, the investment is a good one. But suppose I pose the question of how much I stand to gain by delaying the investment to some later time $t$, say. The return term above is modified to $R_0 e^{\mu t}/(\rho - \mu)$, and if I leave $K$ as is, this new NPV will be strictly greater than the old value: a speculative bubble. I hence have two questions.

Should $K$ be discounted when computing the NPV of the investment at some future time? If so, what would the rationale be? I can understand the time value of money applied to the return, but why should a future investment (i.e. a future cost) be discounted?
If the answer to (i) is in the affirmative, would it make sense to use a different discount rate for $K$ than $\rho$? It seems to me that costs should be discounted differently (more gently) than returns.
To be clear, we are assuming that the nominal investment value is a constant $K$. As such, if we calculate the NPV in the future, that future NPV will rise.
Of course, we do not work with future NPV’s, we are making a decision now, and so we need to discount that future NPV by the discount rate, to get an apples-to-apples comparison.
I will now look at the sub-questions.

An investment $K$ in the future is not the same thing as an investment $K$ now. We can buy a zero coupon bond with maturity $t$, and meet the future outflow of size $K$ with an bond that costs $Ke^{-r_bt}$, where $r_b$ is the mathematical discount rate on the bond. (Quoted yields follow different conventions, so you need to adjust quoted yields for the calculation.)
The reason I would discount future outflows at the bond rate is because that is the mechanism to translate current cash to future cash flows in a “guaranteed” fashion. You could use the project discount rate, but you are assuming reinvestment at the project discount rate, which is awkward if the project is only implemented in the future. The exact answer will depend on conventions for the NPV calculation, but there are pros and cons of each definition.

Which bond to use? If one is cautious, one sticks to “risk-free” sovereign bonds. (Disclaimer: all bonds have risks.) If one feels lucky, one can take credit risk with the investment, raising the bond yield. However, default risk then needs to be taken into account in the NPV analysis.
Since one might not find a zero coupon bond, one needs to calculate a zero coupon curve from observed bond yields. This is non-trivial, but descriptions would be found in most textbooks of fixed income analysis.