Many features used in Structural Health Monitoring strategies are not just highly sensitive to failure mechanisms, but also depend on environmental or operational fluctuations. To prevent incorrect failure uncovering due to these dependencies, damage detection approaches can use robust and temperature-independent features. These indicators can be naturally insensitive to environmental dependencies or artificially made independent. This work explores both options. Cointegration theory is used to remove environmental dependencies from dynamic features to create highly sensitive parameters to detect failure mechanisms: the cointegration residuals. This paper applies the cointegration technique for damage detection of a concrete-masonry tower in Italy. Two regression models are implemented to capture temperature effects: Prophet and Long Short-Term Memory networks. Results demonstrate the advantages and limitations of this methodology for real applications. The authors suggest to combine the cointegration residuals with a secondary temperature-insensitive damage-sensitive set of features, the Cepstral Coefficients, to address the possibility of capturing undetected structural damage.