In recent years, multiterminal Josephson junctions (MTJJs) have emerged as a convenient platform for exploring artificial topology. In these systems, the Andreev bound-state spectrum forms an artificial band structure in synthetic dimensions defined by the superconducting phase differences
between the leads. In particular, Andreev bound states (ABS) can acquire nonzero Chern numbers, which manifest themselves at zero temperature in quantized transconductance between two distinct superconducting leads. However, realistic experiments are performed at finite temperature, which can obscure this quantization. We address this problem theoretically by calculating the finite-temperature transconductance of MTJJs with topological Andreev spectra. We show that finite temperature suppresses the contribution of topological ABS with energies comparable to the temperature, leading to a violation of transconductance quantization. At the same time, we predict that finite-temperature transconductance measurements can reveal topological crossings of ABS at finite energies, which are inaccessible in zero-temperature transconductance measurements.