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How much current can I safely pull from an ESP8266?
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The voltage regulator is only used to generate the 3.3V supply for the ESP. But OP is using the 5V from the input of the regulator to power the LEDs.
At max it is wasting about 0.7W if the 400mA max current is right. But it will still get hot due to the current draw of the ESP. Even at half of the max. rated current this device is probably 30-40 degrees above your ambient temperature.