Can I keep the factory head unit and upgrade the rest of the audio system in a 2010 Honda accord with premium sound?

If the starter is cranking the engine, then the battery's "charge" level is perfectly fine. If the battery were a problem, the starter couldn't crank the engine, but your description makes it sound like it's cranking just not starting immediately when cold. It wouldn't hurt to pull a spark plug or two out of the engine to inspect for excessive wear, especially if the plugs haven't been changed in a long time.

If the plug electrode gaps are obviously wide and out of spec, I'd replace all the plugs and the plug wires attached to them and see if that helps. Additionally, if you're running a thicker motor oil than Honda recommends for the engine (say you're running a 10w30 instead of the 5w20 or 5w30 they designed the engine to use) that could also make it tougher to start when cold. The first number in an oil's grade tells you how the oil behaves at low temperature.

A 10w oil will have a viscosity of about 250cS (centistokes, a unit of viscosity measure) at 70 degrees F. A 5w oil's viscosity at 70 degrees F is approximately 100cS and the difference between the two increases as temperature decreases. Thicker oil at cold starting temp lowers the RPM at which the starter can turn the engine to start it and that can make an engine tougher to start cold in colder weather.

If you're running a 10w oil, you might try switching to a 5w oil and see if that helps too. Good luck.

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