The Roku device does go into a sleep mode when it is not being used. Normal behavior is to press the Home button in order to wake it up.
Some HDMI issues are caused by the Auto-Power Savings setting. Try disabling it under Settings/System/Power/Auto Power Savings/20-min [uncheck to Disable]
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Generally, you press Home to wake Roku device and get to the Home Screen. After you are finished streaming, you press Home again to return to Home Screen (stops streaming and using bandwidth/data). After a period of time, the Roku device will enter into a sleep mode. The Home button again wakes the device and puts you on the Home Screen.
