Here are some of the ROMs that you can try on your Sony Experia P. Make sure you check if the ROM has been tested on the model before you root the device for installation.
1. SlimROM: This ROM works with almost all the android phones. It is frugal version of the default android OS. The ROM also reduces the amount of RAM being used by the apps. So this is the ROM you need if you don't want ROM to consume much resource.
2. Cataclysm: Another frugal ROM with more focus on the performance. I'd say, compared to SlimROM this may consume more RAM but still good on 1GB memory. The current release of the cataclysm was only meant for the Nexus devices, however XDA developers have ported the ROM to other devices. So this option is now open to you.
3. XenonHD: This is more suitable for you if you run games and need more performance from ROM. Most of the ROMs run low when they are on 1GB RAM. This ROM is officially tested on Sony devices. So you can be rest assured that it works with your phone.
4. Paranoid Android ROM is simple and clean UI. Compared to default ROM, it is much easier to navigate. Also it takes less resources compared to Sony's UI and even CyanogenMod mod. It doesn't have much bloatware into it, so it can be easy for you to choose this ROM without much compromising on the UI side.
5. Honami : It was designed for the Xperia Z stock ROM. And it is now being ported to other Sony devices. This ROM looks similar to the MIUI 5 ROM. It doesn't add much bloatware into your device. It has quick navigation like Xeon. Also works good with low RAM such as 1GB.
Avatar: This ROM is basically combo of MIUI and CyanoGenMod. It doesn't add much bloatwaare into your system. It is good for tweaking and for customization. It's also lot quicker on phone than Honami and Paranoid Android.
There are many other frugal ROMs that you can try for performance improvement. Just check if those ROM supports your device. Often some ROMs can be easily ported but some requires tweaking before installing. So I suggest keeping tab on XDA developers forum.
Note: You can install MIUI 7, Oxygen OS 8, CoolUI 8, FlymeOS on XPeria too. However do note that these OS may not be good enough for device that has 1GB RAM. For example, phone with MIUI 7 runs very slow and can't run two apps at the same time. However MIUI 6 maybe good for the Xperia P. I don't suggest installing other high resource hungry ROMs such as Flyme, Oxygen etc.