In this paper, we consider the recoverable robust shortest path problem in acyclic digraphs, employing interval budgeted uncertainty to model uncertain second-stage costs. For the continuous budgeted uncertainty model, we prove that the problem is strongly NP-hard even in layered digraphs. Furthermore, we show that in general acyclic digraphs, the problem cannot be approximated within any constant factor unless $\mathrm{P} = \mathrm{NP}$, nor can it be approximated within a factor of $2^{\log^{1-ε} n}$ for any $ε> 0$ unless $\mathrm{NP} \subseteq \mathrm{DTIME}(n^{\mathrm{poly} \log n})$. For the discrete budgeted uncertainty model, we show that the problem is not approximable unless $\mathrm{P} = \mathrm{NP}$, even in layered digraphs. Finally, we establish that under continuous budgeted uncertainty, the integrality gap of a relaxation allowing a fractional first-stage solution is at least $Ω(\sqrt{n})$.

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