Liquid-liquid phase separation (LLPS) in protein systems is relevant for many phenomena, from protein condensation diseases to subcellular organization to possible pathways toward protein crystallization. Understanding and controlling LLPS in proteins is therefore highly relevant for various areas of (biological) soft matter research. Solutions of the protein bovine serum albumin (BSA) have been shown to have a lower critical solution temperature-LLPS (LCST-LLPS) induceable by multivalent salts. Importantly, the nature of the multivalent cation used influences the LCST-LLPS in such systems. Here, we present a systematic ultrasmall-angle X-ray scattering investigation of the kinetics of LCST-LLPS of BSA in the presence of different mixtures of HoCl3 and LaCl3, resulting in different effective interprotein attraction strengths. We monitor the characteristic length scales xi(t, T-fin) after inducing LLPS by subjecting the respective systems to temperature jumps in their liquid-liquid coexistence regions. With increasing interprotein attraction and increasing T-fin, we observe an increasing deviation from growth law of xi similar to t(1/3) and an increased trend toward arrest. We thus establish a multidimensional method to tune phase transitions in our systems. Our findings help shed light on general questions regarding LLPS and the tunability of its kinetics in both proteins and colloidal systems.