Tailoring the operating voltage to fine-grain temporal changes in the power and performance needs of the workload can effectively enhance power efficiency. Therefore, power-limited computing platforms of today widely deploy integrated (i.e., on-chip) voltage regulation which enables fast fine-grain voltage control. Voltage regulators convert and distribute power from an external energy source to the processor. Unfortunately, power conversion loss is inevitable and projected integrated regulator designs are unlikely to eliminate this loss even asymptotically. Reconfigurable power delivery by selective shut-down, i.e., gating, of distributed on-chip regulators in response to spatio-temporal changes in power demand can sustain operation at the minimum conversion loss. However, even the minimum conversion loss is sizable, and as conversion loss gets dissipated as heat, on-chip regulators can easily cause thermal emergencies due to their small footprint. Although reconfigurable distributed on-chip power delivery is emerging as a new design paradigm to enforce sustained operation at minimum possible power conversion loss, thermal implications have been overlooked at the architectural level. This paper hence provides a thermal characterization. We introduce ThermoGater, an architectural governor for a collection of practical, thermally-aware regulator gating policies to mitigate (if not prevent) regulator-induced thermal emergencies, which also consider potential implications for voltage noise. Practical ThermoGater policies can not only sustain minimum power conversion loss throughout execution effectively, but also keep the maximum temperature (thermal gradient) across chip within 0.6°C (0.3°C) on average in comparison to thermally-optimal oracular regulator gating, while the maximum voltage noise stays within 1.0% of the best case voltage noise profile.