The gauge of a power cord refers to its physical size and electrical current carrying capacity. A higher gauge power cord will have a larger diameter and can carry more current, which may be necessary for high power applications.
For a tube amplifier, the power supply voltage and current must be carefully considered to ensure that they are within the amplifier's specifications. If the power cord is too small, it may cause voltage drops or current surges, which can affect the amplifier's performance and even cause damage. Therefore, when using a tube amplifier, it is recommended to use a power cord that meets the amplifier's power requirements.
In your case, if the amplifier shutdowns occasionally on startup, it may be due to the power cord's ability to handle the initial surge of power when the amplifier is turned on. A higher gauge power cord may be able to handle the initial surge better and prevent such shutdowns. However, before replacing the power cord, it is recommended to check the amplifier's specifications to ensure that the new power cord meets the requirements. It is also recommended to consult with a professional technician for help to ensure the safety and performance of the amplifier.