What is the point of having Excel running without a document open? So that it opens the next document faster? If so, then make sure you have lots of memory (RAM) and then quit using CMD-Q instead of closing the window using CMD-W.
When you have lots of RAM, modern operating systems cache or preserve most or all of the application in memory for when you need it next. This make starting the application next time super quick.
You can see this when you startup your computer. Any large application you start takes a while to come up. If you close it and then re-open it, it is very quick. This is because most of it was still in memory, kept close at hand, so to speak.
But, should you load an application that needs lots of memory, the system may remove Excel from its cache and the next time Excel needs to startup it will be slower…