Final answer:
The best renal replacement therapy for a hemodynamically unstable patient with acute kidney injury is Continuous venovenous hemodialysis (CVVHD), as it allows for slower and more stable management of fluid and wastes compared to conventional hemodialysis.
Step-by-step explanation:
When considering renal replacement therapy for a patient who has acute kidney injury and is hemodynamically unstable, the therapy that their hemodynamic status is most likely to tolerate is Continuous venovenous hemodialysis (CVVHD). Unlike conventional hemodialysis, CVVHD is a gentler process that allows for slower removal of fluid and wastes from the blood, thereby making it a suitable option for patients who are hemodynamically unstable. This method typically requires continuous treatment over 24 hours and is therefore better for managing fluid balance and providing stable waste removal compared to the intermittently applied hemodialysis, which could potentially cause rapid shifts in fluid and electrolyte balance and exacerbate the patient's instability.