Quantum thermometry provides a key capability in temperature estimation, but most existing strategies rely on equilibrium thermometry, where the probe gets thermalized with respect to the temperature to be estimated. This equilibrium paradigm imposes intrinsic limitations: sensitivity is tied to long-time thermalization and often cannot be improved in fast, noisy, or nonstationary settings. On the other hand, the Mpemba effect, a counterintuitive phenomenon where hotter states relax faster than colder ones to a state that is colder than the other two, has mostly been viewed as a thermodynamic anomaly. Here, we establish a fundamental implication of this anomalous effect in quantum thermometry that directional Mpemba inversion is a sufficient condition that guarantees a nonequilibrium enhancement of the Fisher information for temperature estimation. Through explicit analyses of two-level and Λ-level probes coupled to bosonic baths, we show that hotter initializations can outperform both equilibrium strategies and colder initialization over a finite-time window, with a transient Fisher information enhancement of up to approximately a factor of two relative to equilibrium, realizing a metrological Mpemba effect. Our results establish that anomalous relaxation can be a general design principle for nonequilibrium quantum thermometry, enabling ultrafast and nanoscale sensing protocols that exploit, rather than avoid, nonequilibrium dynamics.