Motivated by the increasing use of condition monitoring technology for electrical transformers, this paper deals with the optimal replacement of a system having a hazard function that follows the proportional hazards model with a semi-Markovian covariate process, which we assume is under continuous monitoring. Although the optimality of a threshold replacement policy to minimize the long-run average cost per unit time was established previously in a more general setting, the policy evaluation step in an iterative algorithm to identify optimal threshold values poses computational challenges. To overcome them, we use conditioning to derive an explicit expression of the objective in terms of the set of state-dependent threshold ages for replacement. The iterative algorithm is customized for our model to find the optimal threshold ages. A three-state example illustrates the computational procedure, as well as the effects of different sojourn time distributions of the covariate process on the optimal policy and cost. Numerical examples and sensitivity analysis provide some insights into the suitability of a Markov approximation, and the sources of variability in the cost. The optimization method developed here is much more efficient than the approach that approximates continuous monitoring as periodic, and then optimizes the periodic monitoring parameters.