Monero does make use of SHA-3 [1], though I think it's AES that's the bulk of the time, so if you want to interface with hardware blocks, SHA-3 is probably not worth it. Intel processors (and possibly others too) have circuitry to perform some AES steps, so this already gives you a pretty good speedup over software implementation. Linux does help, generally. I think you misspoke when you said "The processor makes use of the Linux crypto api.". Maybe you mean there is a driver for your SHA-3 hardware module ?
[1] SHA-3 has several posisble configurations, though I think the one that Monero uses is pretty common, but don't quote me on that.