Just my 2 cents:
I’m on a shared hosting with SiteGround, and running GP Premium. My site is blazing fast (first load 1 to 1.5 secs usually, and repeat visits at 500ms to 750ms) after several optimizations, including:
– Using Cache Enabler Plugin from KeyCDN, with the advanced configuration snippet that you add to .htaccess.
– Autoptimize – Aggregage, minify, concatenate CSS/JS, and moved the CSS/JS to Amazon CloudFront
– Removed unwanted Plugins
– Added my site to Cloudflare, and later switched to a WAF solution CDN.
In Pingdom, you should be looking at the DNS seek time, Waiting time as well as the Download time. There is not much you can do about DNS delay, but you can attach your site to CloudFlare for lightning fast (Time to First Byte) TTFB as well as overall delivery time. Cloudflare has an option where you can set it to ignore query strings, and cache the entire site using a Page Rule.
Personally GP is a fantastic light-weight theme, and support is very good. If you can post the Pingdom results page here, someone will take a look.