I want to change my website platform from WordPress to Laravel 5 with new design. If I keep the URLs same as WordPress site, will it affect my current SEO?
Note: I will keep all the pages URL the same.
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Any used framework under the hood has no effect on the SEO. What it matters SEO wise is the front-end (on Site SEO), and the off-site SEO.
Generally any structural, mark-up, content changes have their own effect on SEO. Such changes can be done even with no underlying technology change.
For sure changing framework can't affect off-Site SEO.
For on-site, if after changing your framework you end-up with the exact same mark-up, content, meta and micro data, URLs, speed and anything else that constitute your current site from the SEO perspective, then there will be no effect.
If you have decided to change the framework, this should be a decision based on your project requirements to meet new needs, that another framework can't offer.
You could use this chance to improve your on-site SEO as well. So you may want to plan early and thoroughly, in order to end up with a more robust and solid website, that will bring in new features, functionality and be more optimized for search engines and users alike.
Can you answer all these questions with "yes":
If you can answer these with yes, then no changes will occur. Otherwise there will be changes. These changes might be for the better (e.g. if the new CMS is faster), of not (e.g. it's slower than before).
The question you have to ask yourself is: Is the new CMS better or worse from a SEO point of few, that answer answers your question.