Halt theme activation if attempting to activate on unsupported older php versions.

functions.php

<?php

/**
 * Define the min version you support. Here I use 5.6 as example but change it to match your support version.
 */
define( 'PREFIX_MIN_PHP_VERSION', '5.6' );
/**
 * Attach a function to check php version on the `after_theme_switch` hook.
 */
add_action( 'after_switch_theme', 'prefix_test_for_min_php' );

/**
 * Compair PHP versions against theme defined minimum version required. If min
 * version is not met add an admin notice informing the user and revert to the
 * previously active theme.
 *
 * @return boolean false on failure and nothing on success.
 */
function prefix_test_for_min_php() {

	// Compare PHP versions against our defined min version.
	if ( version_compare( phpversion(), PREFIX_MIN_PHP_VERSION, '<' ) ) {

		// Theme doesn’t meet min php requirements - add admin notice.
		add_action( 'admin_notices', 'prefix_min_php_not_met_notice' );

		// Switch back to previous theme to prevent errors.
		switch_theme( get_option( 'theme_switched' ) );
		return false;

	};
}

/**
 * A partial to generate a dismissable admin notice that should appear if the
 * user attempts to activate this theme on an unsupported php version.
 */
function prefix_min_php_not_met_notice() {
	?>
	<div class="notice notice-error is_dismissable">
		<p>
		<?php esc_html_e( 'You need to update your PHP version to run themename.', 'prefix' ); ?> <br />
		<?php esc_html_e( 'Actual version is:', 'prefix' ); ?> <strong><?php echo phpversion(); // WPCS: XSS ok. ?></strong>, <?php esc_html_e( 'required is', 'prefix' ); ?> <strong><?php echo esc_html( PREFIX_MIN_PHP_VERSION ); ?></strong>
		</p>
	</div>
	<?php
}